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very $0<\rho<\hat\rho_1=\hat\rho_1(\Omega,p)$ problem \eqref{eq:main_prob_U} admits a solution which is a local minimum of
the energy ${\mathcal{E}}$ on ${\mathcal{M}}_\rho$. In particular, $U$ is positive, has Morse index one and the associated solitary w | ave is orbitally
stable.
Furthermore, for every Lipschitz $\Omega$,
\begin{itemize}
\item $\displaystyle 1<p<1+\frac{4}{N} \implies \hat\rho_1\left(\Omega,p\right)
= +\infty$,
\item $\displaystyle p=1+\frac{4}{N} \implies \hat\rho_1\left(\Omega,p\right)
\ |
geq \|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}$,
\item $\displaystyle 1+\frac{4}{N}<p<2^*-1 \implies \hat\rho_1\left(\Omega,p\right)
\geq D_{N,p} \lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}$,
\end{itemize}
where the universal constant $D_{N,p}$ is explicitly w | ritten in terms of $N$ and $p$ in
Section \ref{sec:1const}.
\end{theorem}
\begin{remark}\label{rem:introGS}
Of course, in the subcritical and critical cases, $c_1$ is actually a global minimum.
Furthermore, the lower bound for the supercritical case agrees |
with that of the critical one
since, as shown in Section \ref{sec:1const}, $D_{N,1+4/N} = \|Z_{N,p}\|^2_{L^2({\mathbb{R}}^N)}$
(and $\lambda_1(\Omega)$ is raised to the $0^{\text{th}}$-power). Notice that the estimate for the supercritical case is new al | so in the case $\Omega=B_1$.
\end{remark}
We observe that the exponent of $\lambda_1(\Omega)$ in the supercritical threshold is negative,
therefore such threshold decreases with the size of $\Omega$.
Once the first thresholds have been estimated, we turn |
to the higher ones: by exploiting
the relations between $M_{\alpha,k}$ and $c_k$, we can show that the thresholds obtained
for Morse index one--solutions in Theorem \ref{thm:intro_GS} can be increased, by
considering higher Morse index--solutions, at least | for some exponent.
\begin{proposition}\label{thm:intro_3>1}
For every $\Omega$ and $1<p<2^*-1$,
\[
\hat\rho_3\left(\Omega,p\right) \geq 2 \cdot D_{N,p} \lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}.
\]
\end{proposition}
\begin{remark}
In the critical case |
, the lower bound for $\hat\rho_3$ provided by Proposition \ref{thm:intro_3>1} is twice that for $\hat\rho_1$ obtained in Theorem \ref{thm:intro_GS}.
By continuity, the estimate for $\hat\rho_3$ is larger than that for $\hat\rho_1$ also when
$p$ is supercr | itical, but not too large. To quantify such assertion, we can use Yang's inequality
\cite{MR1894540,MR2262780}, which implies that for every $\Omega$ it holds
\[
\lambda_3(\Omega)\leq \left(1+\frac{N}{4}\right)2^{2/N} \lambda_1(\Omega).
\]
We deduce that $ |
2 \cdot D_{N,p} \lambda_3(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}
\geq D_{N,p} \lambda_1(\Omega)^{\frac{2}{p-1}-\frac{N}{2}}$ whenever
\[
p\leq 1+\frac{4}{N} + \frac{8}{N^2\log_2\left(1+\frac{4}{N}\right)}.
\]
In particular, the physically relevant case $N=3$ | , $p=3$ is covered. Furthermore, if
$N\geq 7$, the above condition holds for every $p<2^*-1$.
\end{remark}
Beyond existence results for
\eqref{eq:main_prob_U}, also multiplicity results can be achieved. A first general
consideration, with this respect, is |
that Theorem \ref{thm:genus_1constr} holds true
also when using the standard Krasnoselskii genus instead of $\gamma$; this allows to
obtain critical points having Morse index bounded from below (see
\cite{MR968487,MR954951,MR991264}), and therefore to obta | in infinitely many solutions,
at least when $\rho$ is less than some threshold. More specifically, we can also
prove the existence of a second solution in the supercritical case, thus extending to
any $\Omega$ the multiplicity result obtained in \cite{MR33 |
18740} for the ball.
Indeed, on the one
hand, in the supercritical case ${\mathcal{E}}_\mu$ is unbounded from below; on the other hand
the solution obtained in Theorem \ref{thm:genus_1constr}, for $k=1$, is a local minimum.
Thus the Mountain Pass Theorem \ | cite{MR0370183} applies on $\mathcal{M}$, and a second solution can be
found for $\mu<\hat\mu_1$, see Proposition \ref{mpcritlev} for further details (and also
Remark \ref{rem:further_crit_lev} for an analogous construction for $k\ge2$).
To conclude thi |
s introduction, let us mention that the explicit lower bounds obtained in
Theorem \ref{thm:intro_GS} can be easily applied in order to gain
much more information also in the case of special domains, as those considered in
Remark \ref{rem:specialdomains}. F | or instance, we can prove then following.
\begin{theorem}\label{pro:symm}
Let $\Omega=B$ be a ball in ${\mathbb{R}}^N$. Then
\[
p<1+\frac{4}{N-1}
\quad\implies\quad
\text{\eqref{eq:main_prob_U} admits a solution for every }\rho>0.
\]
An analogous result ho |
lds when $\Omega=R$ is a rectangle, without further restrictions on
$p<2^*-1$.
\end{theorem}
Therefore our starting problem in $\Omega=B$ can be solved for any mass value also in the critical and
supercritical regime, at least for $p$ smaller than this fur | ther critical exponent $1+4/(N-1) > 1+ 4/N$.
Of course, higher masses require higher Morse index--solutions. In particular, since by \cite{MR3318740}
we know that ${\mathfrak{A}}_1(B,1+4/N) = (0,\|Z_{N,p}\|_{L^2})$, we have that for
larger masses, even tho |
ugh no positive solution exists, nodal solutions with higher
Morse index can be obtained: in such cases \eqref{eq:main_prob_U} admits \emph{nodal ground
states with higher Morse index}.
The paper is structured as follows: in Section \ref{sec:blow-up} we p | erform a blow-up analysis of
solutions with bounded Morse index, in order to prove Theorem \ref{thm:bbd_index}; Section \ref{sec:2const}
is devoted to the analysis of the variational problem with two constraints \eqref{maxmin}
and to the proof of Theorem \ |
ref{thm:genus_2constr};
that of Theorems \ref{thm:genus_1constr}, \ref{thm:intro_GS} and Proposition
\ref{thm:intro_3>1} is developed
in Section \ref{sec:1const}, by means of the variational problem with one constraint
\eqref{infsuplev}; finally, Section
\ | ref{sec:symm} contains the proof of Theorem \ref{pro:symm}.
\textbf{Notation.} We use the standard notation
$\{\varphi_k\}_{k\geq1}$ for a basis of eigenfunctions of the Dirichlet laplacian in $\Omega$,
orthogonal in $H^1_0(\Omega)$ and orthonormal in $ |
L^2(\Omega)$. Such functions are ordered in such
a way that the corresponding eigenvalues $\lambda_k(\Omega)$ satisfy
\[
0<\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\dots,
\]
and $\varphi_1$ is chosen to be positive on $\Omega$. $C_{N,p}$ | denotes the universal constant in
the Gagliardo-Nirenberg inequality \eqref{sobest}, which is achieved (uniquely, up to translations and dilations)
by the positive, radially symmetric function $Z_{N,p}\in H^1({\mathbb{R}}^N)$, with
\[
\|Z_{N,p}\|^2_{L^2({ |
\mathbb{R}}^N)}=\left(\frac{p+1}{2C_{N,p}}\right)^{N/2}.
\]
Finally, $C$ denotes every (positive) constant we need not to specify, whose value may change also within the same formula.
\section{Blow-up analysis of solutions with bounded Morse index}\label{ | sec:blow-up}
Throughout this section we will deal with a sequence $\{(u_n,\mu_n,\lambda_n)\}_n \subset H^1_0(\Omega)\times{\mathbb{R}}^+\times{\mathbb{R}}$ satisfying
\begin{equation}\label{eq:auxiliary_n}
-\Delta u_n+\lambda_n u_n=\mu_n |u_n|^{p-1}u_n,\q |
quad\int_\Omega u_n^2\, dx=1,\qquad \int_\Omega |\nabla u_n|^2\, dx=:\alpha_n.
\end{equation}
To start with, we recall the following result (actually, in \cite{MR3318740}, the result is stated for positive solution, but the proof
does not require such assu | mption).
\begin{lemma}[{\cite[Lemma 2.5]{MR3318740}}]\label{lemma:case_alpha_n_bounded}
Take a sequence $\{(u_n,\mu_n,\lambda_n)\}_n$ as in \eqref{eq:auxiliary_n}. Then
\[
\{\alpha_n\}_n \text{ bounded}
\qquad\implies\qquad
\{\lambda_n\}_n,\,\{\mu_n\}_n\te |
xt{ bounded}.
\]
\end{lemma}
Next we turn to the study of sequences having arbitrarily large $H^1_0$-norm. In particular, we will focus on sequences of solutions having a common upper
bound on the Morse index
\[
m(u_n) = \max\left\{k : \begin{array}{l}
\ex | ists V\subset H^1_0(\Omega),\,\dim(V)= k:\forall v\in V\setminus\{0\}\smallskip\\
\displaystyle\int_\Omega |\nabla v|^2 + \lambda_n v^2 - p\mu_n|u_n|^{p-1}v^2\,dx<0
\end{array}
\right\}.
\]
Throughout this section we will assume that
\begin{equation}\label |
{eq:mainass_secMorse}
\text{the sequence }\{(u_n,\mu_n,\lambda_n)\}_n\text{ satisfies \eqref{eq:auxiliary_n},
with }\alpha_n\to+\infty\text{ and }m(u_n)\leq \bar k,
\end{equation}
for some $\bar k\in{\mathbb{N}}$ not depending on $n$.
\begin{lemma}\label{l | em:lambda_bdd_below}
Let \eqref{eq:mainass_secMorse} hold. Then
\(
\lambda_n \geq -\lambda_{\bar k}(\Omega).
\)
\end{lemma}
\begin{proof}
Assume, to the contrary, that for some $n$ it holds $\lambda_n < -\lambda_{\bar k}(\Omega)$. For any real $t_1,\dots t |
_{\bar k}$
we define
\[
\phi := \sum_{h=1}^{\bar k} t_h \varphi_h.
\]
By denoting $J_{\lambda,\mu}(u)={\mathcal{E}}_\mu(u)+\frac{\lambda}{2}\|u\|_{L^2}^2$, so that
Morse index properties can be written in terms of $J''_{\lambda,\mu}$, we have
\[
\begin{spl | it}
J''_{\lambda_n,\mu_n}(u_n)[u_n,\phi] &= -(p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n\phi,\\
J''_{\lambda_n,\mu_n}(u_n)[\phi,\phi] &= \sum_{h=1}^{\bar k} t_h^2 \int_\Omega \bigl (|\nabla \varphi_h| + \lambda_n \varphi_h^2\bigr )\,dx - p\mu_n\int_\Omega |u_n |
|^{p-1}\phi^2\,dx \\
&\leq \sum_{h=1}^{\bar k} t_h^2(\lambda_{h}(\Omega) + \lambda_n) - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx \leq - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx,
\end{split}
\]
where equality holds if and only if $t_1=\dots=t_{\ba | r k}=0$. As a consequence
\begin{multline*}
J''_{\lambda_n,\mu_n}(u_n)[t_0 u_n+ \phi, t_0 u_n + \phi] \leq -t_0^2 (p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n^2 \\
- 2t_0(p-1)\mu_n\int_\Omega |u_n|^{p-1}u_n\phi \,dx - (p-1)\mu_n\int_\Omega |u_n|^{p-1}\phi^2\,dx. |
\end{multline*}
We deduce that $J''_{\lambda_n,\mu_n}(u_n)$ is negative definite on $\spann\{u_n, \varphi_1,\dots,\varphi_{\bar k}\}$, in contradiction with the bound on the Morse index (note that
$u_n$ cannot be a linear combination of a finite number of | eigenfunctions, otherwise using the equations we would obtain that such eigenfunctions are linearly dependent).
\end{proof}
\begin{lemma}
\label{localblow}
Let \eqref{eq:mainass_secMorse} hold.
Then $\lambda_n\to +\infty$.
\end{lemma}
\begin{proof}
By Lem |
ma \ref{lem:lambda_bdd_below} we have that $\lambda_n$ is bounded below. As a consequence, we can use H\"{o}lder inequality with $\|u_n\|_{L^2}=1$ and \eqref{eq:auxiliary_n} to write
\[
\mu_n\,\|u_n\|^{p-1}_{L^{\infty}}\ge \mu_n \,\|u_n\|^{p+1}_{L^{p+1}}=\ | alpha_n+\lambda_n
\rightarrow +\infty.
\]
Let us define
\begin{equation}
\label{equn}
U_n: =\mu_n^{\frac{1}{p-1}}\,u_n,
\quad\text{ so that }
-\Delta U_n+\lambda_n U_n=|U_n|^{p-1}U_n\quad \text{in}\,\,\Omega,\quad
U|_{\partial\Omega}=0.
\end{equation}
Pick |
$P_n\in\Omega$ such that $|U_n(P_n)|=\|U_n\|_{L^{\infty}(\Omega)}$ and set
\begin{equation}
\label{tildepsn}
\tilde\varepsilon_n: =|U_n(P_n)|^{-\frac{p-1}{2}}=\frac{1}{\sqrt{\mu_n\,\|u_n\|^{p-1}_{L^{\infty}}}}\longrightarrow 0
\end{equation}
Hence, $|U_n( | P_n)|\to+\infty$; moreover, as $P_n$ is a point of positive maximum or of negative minimum, we have
\begin{equation}
\nonumber
0\le \frac{-\Delta U_n(P_n)}{U_n(P_n)}=|U_n(P_n)|^{p-1}-\lambda_n\,.
\end{equation}
Thus $\lambda_n|U_n(P_n)|^{1-p}\le 1$, and si |
nce $\lambda_n$ is bounded from below, we conclude
\begin{equation}
\label{limtildelam}
\frac{\lambda_n}{|U_n(P_n)|^{p-1}}\longrightarrow \tilde\lambda\in [0,1].
\end{equation}
Now, we are left to prove that $\tilde\lambda>0$. Let us define
\begin{equation | }
\label{tildeVn}
\tilde V_n(y)=\tilde\varepsilon_n^{\frac{2}{p-1}}\, U_n(\tilde\varepsilon_n\,y+P_n),\quad\quad y\in \tilde\Omega_n
:=\big (\Omega-P_n\big )/\tilde\varepsilon_n,
\end{equation}
and let $d_n := d(P_n,\partial\Omega)$; we have, up to subsequ |
ences,
\[
\frac{\tilde\varepsilon_n}{d_n}\longrightarrow L\in [0,+\infty]
\qquad\text{and}\qquad
\tilde\Omega_n\rightarrow\left\{
\begin{array}{ll}
{\mathbb{R}}^n, & \text{if $L=0$;} \\
H, & \text{if $L>0 | $,}
\end{array}
\right.
\]
where $H$ is a half-space such that $0\in \overline H$ and $d(0,\partial H)=1/L$.
The function $\tilde V_n$ satisfies
\begin{equation}
\nonumber
\left\{
\begin{array}{ll}
-\Delta \tilde V_n+\la |
mbda_n\,\tilde \varepsilon_n^2\,\tilde V_n=|\tilde V_n|^{p-1}\tilde V_n, & \hbox{in}\,\, \tilde\Omega_n;\\
|\tilde V_n|\le |\tilde V_n(0)|=1, & \hbox{in}\,\, \tilde\Omega_n;\\
\tilde V_n=0, & \hbox{on}\,\, \partial\tilde \Omega_n.
\end{array}
\ri | ght.
\end{equation}
From \eqref{tildepsn} and \eqref{limtildelam} we get $\tilde\varepsilon_n^2\,\lambda_n\rightarrow \tilde\lambda$; hence, by elliptic regularity and up to a further subsequence, $\tilde V_n\rightarrow \tilde V$ in $\mathcal{C}^1_{{\mathr |
m{loc}}}(\overline H)$ where $\tilde V$ solves
\begin{equation}
\label{limprob1}
\left\{
\begin{array}{ll}
-\Delta \tilde V+\tilde\lambda\,\tilde V=|\tilde V|^{p-1}\tilde V, & \hbox{in}\,\, H;\\
|\tilde V|\le |\tilde V(0)|=1, & \hbox{in}\,\, H;\\ |
\tilde V=0, & \hbox{on}\,\, \partial H.
\end{array}
\right.
\end{equation}
Since $\sup_n m(U_n)\leq \bar k$ (as a solution to \eqref{equn}), one can show as in Theorem $3.1$ of \cite{MR2825606} that $m(\tilde V)\leq \bar k$. In particular, $\tilde V |
$ is stable outside a compact set (see Definition $2.1$ in \cite{MR2825606}) so that, by Theorem $2.3$ and Remark $2.4$ of \cite{MR2825606}, we have
$$\tilde V(x)\rightarrow 0 \quad\quad \text{as} \quad\quad |x|\rightarrow +\infty.$$
Moreover, since $\til | de V$ is not trivial, we also have that $\tilde\lambda>0$. For, if $\tilde\lambda=0$ the function
$\tilde V$ would be a solution of the Lane-Emden equation
$-\Delta u=|u|^{p-1}u$ either in ${\mathbb{R}}^n$ or in $H$. In both cases, $\tilde V$ would contra |
dict Theorems $2$ and $9$ of \cite{MR2322150}, being non trivial and stable outside a compact set. Thus, $\tilde\lambda >0$ and by \eqref{limtildelam} we conclude $\lambda_n\rightarrow +\infty$.
\end{proof}
\begin{remark}\label{rem4}
We stress that the sca | ling argument in Lemma \ref{localblow}, leading to the limit problem \eqref{limprob1} (with $\tilde\lambda>0$), can be repeated also near points of \emph{local} extremum.
More precisely, let $Q_n$ be such that $|U_n(Q_n)|\to +\infty$ and
$$
|U_n(Q_n)|=\max |
_{\Omega\cap B_{R_n\tilde\varepsilon_n}(Q_n)}U_n,
$$
for some $R_n\to +\infty$. Then the above procedure can be repeated by replacing $P_n$ with $Q_n$ in definition \eqref{tildepsn}.
\end{remark}
The local description of the asymptotic behaviour of the so | lutions $U_n$ to \eqref{equn} with bounded Morse index can be carried out more conveniently by defining
the sequence (see \cite[Theorem $3.1$]{MR2825606})
\begin{equation}
\label{defVn}
V_n(y)=\varepsilon_n^{\frac{2}{p-1}}\, U_n(\varepsilon_n\,y+P_n),\quad |
y\in \Omega_n
:=\frac{\Omega-P_n}{\varepsilon_n},
\end{equation}
where $P_n$ is defined before \eqref{tildepsn}, and $\varepsilon_n=\frac{1}{\sqrt{\lambda_n}}\to 0$.
Then, $V_n$ satisfies
\begin{equation}
\nonumber
\left\{
\begin{array}{ll}
-\Delta | V_n+ V_n=| V_n|^{p-1} V_n, & \hbox{in}\,\,\Omega_n;\\
|V_n|\le | V_n(0)|=\big ({\varepsilon_n/\tilde\varepsilon_n}\big )^{\frac{2}{p-1}}\rightarrow
\tilde\lambda^{-\frac{1}{p-1}}, & \hbox{in}\,\, \Omega_n;\\
V_n=0, & \hbox{on}\,\, \partial\Omega_n. |
\end{array}
\right.
\end{equation}
As before, we have (up to a subsequence) $V_n\rightarrow V$ in $\mathcal{C}^1_{\mathrm{loc}}(\overline H)$ where $H$ is either ${\mathbb{R}}^N$ or a half space and $V$ solves
\begin{equation}
\label{limprob2}
\left\{
| \begin{array}{ll}
-\Delta V+ V=| V|^{p-1} V, & \hbox{in}\,\, H;\\
|V|\le | V(0)|=\tilde\lambda^{-\frac{1}{p-1}}, & \hbox{in}\,\, H;\\
V=0, & \hbox{on}\,\, \partial H.
\end{array}
\right.
\end{equation}
By recalling the discussion followin |
g \eqref{limprob1} we also have $m(V)<+\infty$.
We collect some well known property of such a $V$ in the following result.
\begin{theorem}[\cite{MR688279,MR2825606,MR2322150,MR2785899}]\label{thm:unif_est_Farina}
Let $V$ be a classical solution to \eqref{l | improb2} such that $m(V)\leq\bar k$. Then:
\begin{enumerate}
\item $H={\mathbb{R}}^N$;
\item $V(x)\to 0$ as $|x|\rightarrow +\infty$, $V \in H^1({\mathbb{R}}^N)\cap L^{p+1}({\mathbb{R}}^N)$;
\item there exist $C$ only depending on $\bar k$ (and not on $ |
V$) such that
\[
\|V\|_{L^{\infty}} + \|\nabla V\|_{L^{\infty}}<C.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
Claim 2 follows from Theorem 2.3 and Remark 2.4 of \cite{MR2825606}, see also \cite[Remark 1.4]{MR688279}. As a consequence,
Theorem 1.1 of | \cite[Remark 1.4]{MR688279} readily applies, providing claim 1 ($V$ is not trivial as $V(0)>0$). On the other hand, the $L^\infty$ estimates in claim 3.
are proved in Theorem 1.9 of \cite{MR2785899}.
\end{proof}
\begin{corollary}
\label{distpnbound}
If th |
e sequence $\{U_n\}$ of solutions to \eqref{equn} has uniformly bounded Morse index, and if $P_n\in\Omega$ is such that $|U_n(P_n)|=\|U_n\|_{L^{\infty}(\Omega)}\to+\infty$, then
$$
\sqrt{\lambda_n}\,d(P_n,\partial\Omega)\rightarrow +\infty,\qquad\text{whe | re }\frac{\lambda_n}{|U_n(P_n)|^{p-1}}\to\tilde\lambda\in(0,1].
$$
\end{corollary}
\begin{remark}
Recall that $Z_{N,p}$, the unique positive solution to $-\Delta u+ u=| u|^{p-1} u$ in ${\mathbb{R}}^N$, has Morse index $1$ \cite{MR969899}; then, if $V$ sol |
ves \eqref{limprob2} in ${\mathbb{R}}^N$
and $1<m(V)<+\infty$, then $V$ is necessarily sign-changing.
\end{remark}
Following the same pattern as in \cite{MR2825606}, we now analyze the global behaviour of a sequence $\{U_n\}$ of solutions to \eqref{equn} | for $\lambda_n\to +\infty$, assuming that
\(
\lim_{n\to +\infty} m(U_n)\leq\bar k<\infty.
\)
By the previous discussion, if $P^1_n$ is a sequence of points such that $|U_n(P^1_n)|=\|U_n\|_{L^{\infty}(\Omega)}$, we have
$|U_n(P^1_n)|\rightarrow +\infty$ an |
d ${\lambda_n}\,d(P^1_n,\partial\Omega)^2\rightarrow +\infty$. We now look for other possible sequences of (local) extremum points $P^i_n$, $i=2,3,..$, along which $|U_n|$ goes to infinity. For any $R>0$, consider the quantity
\begin{equation}
\nonumber
h_ | 1(R)=\limsup_{n\to +\infty} \Bigl (\lambda_n^{-\frac{1}{p-1}}\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \Bigr ).
\end{equation}
We will prove that if $h_1(R)$ is \emph{not vanishing} for large $R$, then there exists a 'blow-up' sequence $P^2_n$ for $ |
u_n$, 'disjoint' from $P^1_n$. Indeed, let us suppose that
\begin{equation}
\nonumber
\limsup_{R\to +\infty} h_1(R)=4\delta>0.
\end{equation}
Hence, up to a subsequence and for arbitrarily large $R$, we have
\begin{equation}
\label{ass1}
\lambda_n^{-\frac{ | 1}{p-1}}\max_{|x-P^1_n|\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \ge 2\delta.
\end{equation}
Since $U_n$ vanishes on $\partial\Omega$, there exists
$P^2_n\in\Omega\backslash B_{R\,\lambda_n^{-1/2}}(P_n^1)$
such that
\begin{equation}
\label{pn2}
|U_n(P_n^2)|=\max_{ |
|x-P^1_n|\ge R\,\lambda_n^{-1/2}}|U_n(x)|.
\end{equation}
Clearly, assumption \eqref{ass1} implies that $|U_n(P_n^2)|\rightarrow +\infty$. We first prove that the sequences $P_n^1$ and $P_n^2$ are far away each other.
\begin{lemma}
\label{disj}
Take $R$ su | ch that \eqref{ass1} holds, and let $P_n^2$ be defined as in \eqref{pn2}; then
\begin{equation}
\label{limp1p2}
\lambda_n^{1/2}|P_n^2-P^1_n|\rightarrow +\infty
\end{equation}
as $n\to \infty$.
\end{lemma}
\begin{proof}
Assuming the contrary one would get, |
up to a subsequence
\[
\lambda_n^{1/2}|P_n^2-P^1_n|\rightarrow R'\ge R.
\]
Let us now recall
that by \eqref{defVn} and the subsequent discussion, we have:
\begin{equation}
\label{limblowseq}
\lambda_n^{-\frac{1}{p-1}}\, U_n(\lambda_n^{-1/2}\,y+P^1_n) =: V^ | 1_n(y)\rightarrow V(y)\quad \textrm{in}
\,\, \mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^N)
\end{equation}
as $n\to +\infty$. Then, up to subsequences,
\begin{equation}
\nonumber
\lambda_n^{-\frac{1}{p-1}}\,|U_n(P_n^2)|=\big |V^1_n\bigr(\lambda_n^{1/2}(P_n |
^2-P^1_n)\bigl)\big|
\rightarrow \big |V(y')\big |,\quad |y'|=R'\ge R.
\end{equation}
Since $V$ is vanishing for $|y|\to +\infty$, one can choose $R$ such that
$|V(y)|\le\delta$ for every $ |y|\ge R$.
But this contradicts \eqref{ass1}.
\end{proof}
Furtherm | ore, we also have that the blow-up points stay far away from the boundary.
\begin{lemma}
\label{distbd}
Assume \eqref{ass1} and let $P_n^2$ be defined as in \eqref{pn2}; then
\begin{equation}
\label{distp2nbound}
\sqrt{\lambda_n}\,d(P^2_n,\partial\Omega)\r |
ightarrow +\infty
\end{equation}
as $n\to \infty$. Moreover,
\begin{equation}
\label{maxp2nball}
|U_n(P_n^2)|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^2_n)}|U_n|
\end{equation}
for some $R_n\to +\infty$.
\end{lemma}
\begin{proof}
Let us set
\begin{equati | on}
\nonumber
\tilde\varepsilon^2_n: =|U_n(P^2_n)|^{-\frac{p-1}{2}}\quad \mathrm{and}
\quad R_n^{(2)}:=\frac{1}{2}\,\frac{|P_n^2-P_n^1|}{\tilde\varepsilon^2_n}.
\end{equation}
Clearly, $\tilde\varepsilon^2_n\rightarrow 0$; moreover, by \eqref{ass1} and \eq |
ref{pn2},
$\tilde\varepsilon^2_n\le (2\delta)^{-\frac{p-1}{2}}\lambda_n^{-1/2}$, so that
$$R^{(2)}_n\ge \frac{(2\delta)^{\frac{p-1}{2}}}{2}\,\lambda_n^{1/2}\,{|P_n^2-P_n^1|}
\rightarrow +\infty, $$
as $n\to +\infty$ by Lemma \ref{disj}. We claim that this | implies
\begin{equation}
\label{maxp2n}
|U_n(P_n^2)|=\max_{\Omega\cap B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)}| U_n|.
\end{equation}
For, if $x\in B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)$, by \eqref{limp1p2} we would have
$$|x-P^1_n|\ge |P^2_n-P^1_n|- |
|x-P^2_n|\ge \frac{1}{2}\,|P^2_n-P^1_n|\ge R\,\lambda_n^{-1/2},$$
for arbitrarily large $R$. This means that
$$\Omega\cap B_{R^{(2)}_n\tilde\varepsilon^2_n}(P^2_n)\subset \Omega\backslash B_{R\,\lambda_n^{-1/2}}(P^1_n).$$
Then, the claim follows. Now, by r | ecalling Remark \ref{rem4}, we can apply to $U_n$ satisfying \eqref{maxp2n} the same scaling arguments as in the proof of Lemma \ref{localblow}, so that we conclude
$$
0< \lim_{n\to +\infty}\tilde\varepsilon_n^2\,\sqrt{\lambda_n}.
$$
Hence, \eqref{maxp2nb |
all} holds by defining $R_n=R^{(2)}_n\tilde\varepsilon_n^2\,\sqrt{\lambda_n}$,
and \eqref{distp2nbound} follows by Corollary \ref{distpnbound}.
\end{proof}
We can now iterate the previous arguments: let us define, for $k\ge 1$,
\begin{equation}
\label{d | efhn}
h_k(R)=\limsup_{n\to +\infty} \Bigl (\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \Bigr ),
\end{equation}
where
$$d_{n,k}(x): =\min\{|x-P^i_n|\,:\, i=1,...,k\}$$
and the sequences $P^i_n$ are such that
\begin{equation}
|
\nonumber
\sqrt{\lambda_n}\,d(P^i_n,\partial\Omega)\rightarrow +\infty;\quad \lambda_n^{1/2}|P_n^i-P^j_n|\rightarrow +\infty,\quad\quad i,j=1,...,k,\quad i\neq j
\end{equation}
as $n\to +\infty$.
Assume that
$$\limsup_{n\to +\infty} h_k(R)=4\delta>0.$$
As | before, up to a subsequence and for arbitrarily large $R$, we have
\begin{equation}
\label{assk}
\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \ge 2\delta
\end{equation}
and there exist $P^{k+1}_n$ so that
\begin{equation}
\no |
number
|U_n(P_n^{k+1})|=\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}|U_n(x)|
\end{equation}
with $\lim_{n\to +\infty}|U_n(P_n^{k+1})|=+\infty$.
Moreover, as in Lemma \ref{disj} we deduce that, for every $i=1,...,k$
\begin{equation}
\label{limblowseqi}
\lambda_ | n^{-\frac{1}{p-1}}\,U_n(\lambda_n^{-1/2}\,y+P^i_n): = V^i_n(y)\rightarrow V^i(y)\quad \textrm{in}
\,\, \mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^N)
\end{equation}
as $n\to +\infty$; hence, by \eqref{assk} and again from the vanishing of $V$ at infinity, |
we conclude that
\begin{equation}
\lambda_n^{1/2}|P_n^{k+1}-P^i_n|\rightarrow +\infty
\end{equation}
as $n\to \infty$, for every $i=1,...,k$. Setting now
\begin{equation}
\nonumber
\tilde\varepsilon^{k+1}_n: =|U_n(P^{k+1}_n)|^{-\frac{p-1}{2}}\quad \mathrm{ | and}
\quad R_n^{(k+1)}:=\frac{1}{2}\,\frac{d_{n,k}(P^{k+1}_n)}{\tilde\varepsilon^{k+1}_n}
\end{equation}
we still have $\tilde\varepsilon^{k+1}_n\to 0$ and, by \eqref{assk}, $R_n^{(k+1)} \to +\infty$ as $n\to \infty$ (see Lemma \ref{distbd}). Then, by th |
e same arguments as in Lemma \ref{distbd}, we get
\begin{equation}
\label{maxpkn}
|U_n(P_n^{k+1})|=\max_{\Omega\cap B_{R^{(k+1)}_n\tilde\varepsilon^{k+1}_n}(P^{k+1}_n)} |u_n|\,,
\end{equation}
and furthermore
$$ \lim_{n\to +\infty}\tilde\varepsilon_n^{k+1} | \,\sqrt{\lambda_n}>0\,,$$
so that by defining $R_n=:R^{(k+1)}_n\tilde\varepsilon_n^{k+1}\,\sqrt{\lambda_n}\rightarrow +\infty$
we have
\begin{equation}
\label{maxpknball}
|U_n(P_n^{k+1})|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^{k+1}_n)}| U_n|.
\end{equ |
ation}
Now, by the same arguments as in \cite{MR2825606}, it turns out that the iterative procedure must stop after \emph{at most} $\bar k-1$ steps, where
$\bar k =\lim_{n\to +\infty} m(u_n)$. Thus, we have proved:
\begin{proposition}
\label{glob1}
Let $\{ | U_n\}_n$ be a solution sequence to \eqref{equn} such that $\lambda_n\to+\infty$
and $m(U_n)\leq\bar k$.
Then, up to a subsequence, there exist $P_n^1,...,P_n^k$, with $k\le \bar k$ such that
\begin{equation}
\label{limpin}
\sqrt{\lambda_n}\,d(P^i_n,\partia |
l\Omega)\rightarrow +\infty;\quad \lambda_n^{1/2}|P_n^i-P^j_n|\rightarrow +\infty,\quad\quad i,j=1,...,k,\quad i\neq j
\end{equation}
as $n\to +\infty$ and
\begin{equation}
\nonumber
|U_n(P_n^{i})|=\max_{\Omega\cap B_{R_n\lambda^{-1/2}_n}(P^{i}_n)}|U_n|,\q | uad i=1,...,k,
\end{equation}
for some $R_n\to +\infty$ as $n\to +\infty$. Finally,
\begin{equation}
\label{limhr0}
\lim_{R\to +\infty} h_k(R)=0
\end{equation}
where $h_k(R)$ is given by \eqref{defhn}.
\end{proposition}
We now show that the sequence $U_n$ |
decays exponentially away from the blow-up points.
\begin{proposition}
\label{glob2}
Let $\{U_n\}_n$ satisfy the assumptions of Proposition \ref{glob1}. Then, there exist
$P_n^1,...,P_n^k$ and
positive constants $C$, $\gamma$, such that
\begin{equation}
\l | abel{stimglob}
|U_n(x)|\le C\lambda^{\frac{1}{p-1}}_n \sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\,,\quad\quad \forall \,x\in\Omega,\quad n\in{\mathbb{N}}\,.
\end{equation}
\end{proposition}
\begin{proof}
By \eqref{limhr0}, for large $R>0$ and $n>n_0( |
R)$ it holds
\begin{equation}
\nonumber
\lambda_n^{-\frac{1}{p-1}}\max_{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}}
|U_n(x)| \le \Bigr (\frac{1}{2p} \Bigl )^{\frac{1}{p-1} }
\end{equation}
Then, for $n>n_0(R)$ and for $x\in \{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}\}$, | we have
\begin{equation}
\nonumber
a_n(x): = \lambda_n-p |U_n(x)|^{p-1}\ge \lambda_n-\frac{\lambda_n}{2}=\frac{\lambda_n}{2}
\end{equation}
We stress that the linear operator
\begin{equation}
\nonumber
L_n: =-\Delta + a_n(x)
\end{equation}
comes from the l |
inearization of equation \eqref{equn} at $U_n$; let us compute this operator on the functions
$$\phi^i_n(x)=e^{-\gamma\sqrt{\lambda_n}\,|x-P^i_n|}\,,\quad\quad \gamma>0,\quad\quad i=1,...,k$$
in $\{d_{n,k}(x)\ge R\,\lambda_n^{-1/2}\}$. We obtain:
$$L_n \ph | i^i_n(x)=\lambda_n\phi^i_n(x)\Bigr [-\gamma^2+(N-1)\frac{\gamma}{\sqrt{\lambda_n}\,|x-P^i_n|}
+\frac{a_n(x)}{\lambda_n}\Bigl ]\ge \lambda_n\phi^i_n(x)\bigr [-\gamma^2+1/2\bigl ]\ge 0$$
for $n$ large, provided $0<\gamma\le 1/\sqrt 2$. Moreover, for $|x-P^i_ |
n|=R\lambda_n^{-1/2}$, $ i=1,...,k,$ and $R$ large we have
$$
e^{\gamma R}\phi^i_n(x)-\lambda_n^{-\frac{1}{p-1}}|U_n(x)|=
1-\lambda_n^{-\frac{1}{p-1}}|U_n(x)|>0
$$
as $n\to +\infty$, by \eqref{limblowseq}. Note further that
$$\{x:d_{n,k}(x)= R\,\lambda_n^{ | -1/2}\} = \bigcup_{i=1}^k \partial B_{R\,\lambda_n^{-1/2}}(P_n^i)
\subset \Omega$$
for large enough $n$. Then, by defining
\begin{equation}
\nonumber
\phi_n: = e^{\gamma R}\lambda_n^{\frac{1}{p-1}}\sum_{i=1}^k \,\phi^i_n
\end{equation}
we have
$$\phi_n(x)- |
|U_n(x)|\ge 0\quad\quad \mathrm{on}\quad\quad\{d_{n,k}(x)= R\,\lambda_n^{-1/2}\}\cup\partial\Omega$$
and
\begin{equation}
\nonumber
L_n(\phi_n-|U_n|)\ge -L_n\,|U_n|=\Delta \,|U_n|-\lambda_n\,|U_n|+p|U_n|^p\ge (p-1)\,|U_n|^p\ge 0
\end{equation}
in $\Omega\b | ackslash \{d_{n,k}(x)\le R\,\lambda_n^{-1/2}\}$. Then (for $R$ large and $n\ge n_0(R)$)
we obtain $|U_n|\le \phi_n$ in the same set, by the minimum principle. Moreover, since by \eqref{limtildelam}
$$|U_n(x)|\le\|U_n\|_{L^{\infty}(\Omega)}=|U_n(P^1_n)|\le |
C \lambda_n^{\frac{1}{p-1}}$$
for some $C>0$, we also have, in $\{d_{n,k}(x)\le R\,\lambda_n^{-1/2}\}$,
$$
|U_n(x)|\le\|U_n(x)\|_{L^{\infty}(\Omega)}=|U_n(P^1_n)|\le C e^{\gamma R}\lambda_n^{\frac{1}{p-1}}\sum_{i=1}^k e^{-\gamma\sqrt{\lambda_n}|x-P_n^i | |}.
$$
Then, possibly by choosing a larger $C$, estimate \eqref{stimglob} follows for every $n$.
\end{proof}
We now exploit the previous results to show that suitable rescalings of the solutions to \eqref{eq:auxiliary_n} converge (locally) to some bounded |
solution $V$ of
\begin{equation}
\label{eqV}
-\Delta V+ V=| V|^{p-1} V
\end{equation}
in ${\mathbb{R}}^N$.
\begin{lemma}
\label{lemlim1}
Let \eqref{eq:mainass_secMorse} hold.
Then $|u_n|$ admits $k\le \bar k$ local maxima $P_n^1,...,P_n^k$ in $\Omega$ su | ch that, defining
\begin{equation}
u_{i,n}(x)= \Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{1}{p-1}}u_n \bigr (\frac{x}
{\sqrt {\lambda_n}}+P_n^i\bigl ),\quad\quad x\in \Omega_{n,i}:=\sqrt{\lambda_n}\bigr (\Omega-P_n^i \bigl ),
\end{equation}
it results, |
up to a subsequence,
\begin{equation}
u_{i,n}(x)\rightarrow V_i\quad\quad \mathrm{in}\,\,\mathcal{C}^1_{{\mathrm{loc}}}({\mathbb{R}}^n)\quad \mathrm{as}\,\,n\to +\infty,\quad \forall\,\,i=1,2,...,k,
\end{equation}
where $V_i$ is a bounded solution of \eqre | f{eqV} with $m(V_i)\le \bar{k}$.
\noindent As a consequence, for every $q\ge 1$,
\begin{equation}
\label{convlq}
\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx\rightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n} |
|V_i|^q\,dx\quad\quad \mathrm{as}\,\, n\to +\infty.
\end{equation}
\end{lemma}
\begin{proof}
By Lemma \ref{localblow} we have $\lambda_n\to +\infty$; then, the first part of the lemma follows by definition \eqref{equn}, by \eqref{limblowseqi} and by Propos | ition \ref{glob1}; by the same proposition
and by Proposition \ref{glob2} we also have that the local maxima $P^i_n$ satisfies \eqref{limpin} and that the pointwise estimate
\begin{equation}
\label{stimglobvn}
|u_n(x)|\le C\Bigl (\frac{\lambda_n}{\mu_n}\Bi |
gr )^{\frac{1}{p-1}} \sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\,,\quad\quad \forall \,x\in\Omega,\quad n\in{\mathbb{N}}\,.
\end{equation}
holds. Let us fix $R>0$ and set $r_n=R/\sqrt{\lambda_n}$; for large enough $n$, \eqref{limpin} implies
$$B_{r_n | }(P^i_n)\subset \Omega,\quad\quad B_{r_n}(P^i_n)\cap B_{r_n}(P^j_n)=\emptyset, \quad i\neq j.$$
Then we obtain
\[
\begin{array}{cl}
&\left |\left ( \frac{\mu_n}{\lambda_n}\right)^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-
\sum_{j=1}^{k}\int_ |
{B_R(0)}|u_{j,n}|^q\,dx\,\,\right |
\smallskip\\&
=\left ( \frac{\mu_n}{\lambda_n}\right )^{\frac{q}{p-1}}\lambda_n^{N/2}\left |\int_{\Omega} |u_n|^q \,dx-
\sum_{j=1}^{k}\int_{B_{r_n}(P^j_n)}|u_{n}|^q\,dx\,\,\right |
\smallskip\\
&=\left ( \frac{\mu_n}{\la | mbda_n}\right )^{\frac{q}{p-1}}\lambda_n^{N/2}
\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} |u_n|^q \,dx
\le C^q\lambda_n^{N/2}
\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} \left |\sum_{i=1}^ke^{-\gamma\sqrt{\lambda_n}|x-P_n^i|}\right |
|^q \,dx
\smallskip\\
&\le C^qk^{q-1}\lambda_n^{N/2} \sum_{i=1}^k
\int_{\Omega\backslash \bigcup_{j=1}^k\,B_{r_n}(P^j_n)} e^{-q\gamma\sqrt{\lambda_n}|x-P_n^i|} \,dx
\smallskip\\
&\le C^qk^{q-1}\lambda_n^{N/2} \sum_{i=1}^k
\int_{{\mathbb{R}}^N\backslash \, | B_{r_n}(P^i_n)} e^{-q\gamma\sqrt{\lambda_n}|x-P_n^i|} \,dx
\smallskip\\
&\le (Ck)^{q}\sum_{i=1}^k
\int_{{\mathbb{R}}^N\backslash \,B_{R}(0)} e^{-q\gamma\,|y|} \,dy\le C_1 \,e^{-C_2 R},
\end{array}
\]
for some positive $C_1$, $C_2$. Letting $n\to +\infty$ w |
e have, up to subsequences,
\begin{multline*}
\Bigg |\lim_{n\to +\infty}\Bigl ( \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-
\sum_{i=1}^{k}\int_{B_R(0)}|V_i|^q\,dx\,\,\Bigg |
\\
=\lim_{n\to +\infty}\Bigg |\Bigl ( | \frac{\mu_n}{\lambda_n}\Bigr )^{\frac{q}{p-1}}\lambda_n^{N/2}\int_{\Omega} |u_n|^q \,dx-
\sum_{i=1}^{k}\int_{B_R(0)}|u_{i,n}|^q\,dx\,\,\Bigg |\le C_1 \,e^{-C_2 R}.
\end{multline*}
Then, \eqref{convlq} follows by taking $R\to +\infty$.
\end{proof}
The prev |
ious lemma allows us to gain some information on the asymptotic behavior of the sequences $\lambda_n$, $\mu_n$ and $\|u_n\|_{L^{p+1}(\Omega)}$. We first provide some bounds for the solutions of the limit problem \eqref{eqV} which will be useful in the sequ | el.
\begin{lemma}
\label{boundbelow}
Let $V_i$, $i=1,\dots,k$ be as in Lemma \ref{lemlim1} (so that $m(V_i)\leq\bar k$). There exists a constant $C$, only depending on the full sequence $\{u_n\}_n$ and not on $V_i$
(and on the particular associated subsequ |
ence), such that
\[
\|V_i\|_{H^1}^2 = \|V_i\|_{L^{p+1}}^{p+1} \leq C.
\]
Furthermore, if also $m(V_i)\geq2$ (or, equivalently, if $V_i$ changes sign)
the following estimates hold:
\begin{equation}
\label{uppstiml2}
\|V_i\|^{p+1}_{L^{p+1}}> 2\,\|Z\|^{{p+1}} | _{L^{{p+1}}},\qquad
\|V_i\|^2_{L^2}> 2\,\|Z\|^{2}_{L^{2}},
\end{equation}
where $Z\equiv Z_{N,p}$ is the unique positive solution to \eqref{eqV}.
\end{lemma}
\begin{proof}
To prove the bounds from above we claim that there exists $\bar R>0$, not depending |
on $i$, such that $V_i$ is stable outside $\overline{B_{\bar R}}$. Then
the desired estimate will follow, since
\[
\|V_i\|^{p+1}_{L^{p+1}} = \int_{B_{\bar R}} |V_i|^{p+1} + \int_{{\mathbb{R}}^N\setminus B_{\bar R}} |V_i|^{p+1},
\]
where the first term is u | niformly bounded by Theorem \ref{thm:unif_est_Farina}, while the second one can be estimated in an uniform way
by reasoning as in the proof of \cite[Theorem 2.3]{MR2825606}. To prove the claim, recalling \eqref{defhn} and \eqref{limhr0}, let
$\bar R$ be su |
ch that
\[
h_k(\bar R) \leq \left(\frac{1}{p}\right)^{1/(p-1)}.
\]
Then $|V_i(x)|^{p-1}\leq 1/p $ on ${\mathbb{R}}^N\setminus B_{\bar R}$ and thus, for any $\psi\in C^\infty_0({\mathbb{R}}^N)$, $\psi\equiv0$ in $B_{\bar R}$, it holds
\[
\int_{{\mathbb{R}}^ | N} |\nabla \psi|^2 + \psi^2 - p|V_i|^{p-1}\psi^2\,dx \geq \left( 1 - p \|V_i\|^{p-1}_{L^{\infty}({\mathbb{R}}^N\setminus B_{\bar R})}\right)\int_{{\mathbb{R}}^N} \psi^2 \geq 0.
\]
Hence $V_i$ is stable outside $B_{\bar R}$, and the first part of the lemma |
follows.
On the other hand, if $V_i$ is a sign-changing solution to \eqref{eqV}, the associated energy functional
\begin{equation}
\nonumber
E(V_i)= \frac{1}{2}\|\nabla V_i\|^2_{L^2}+\frac{1}{2}\|V_i\|^2_{L^2}-\frac{1}{p+1}\|V_i\|^{p+1}_{L^{p+1}}
\end{e | quation}
satisfies the following \emph{energy doubling property} (see \cite{MR2263672}):
$$E(V_i)>2\,E(Z)$$
On the other hand, by using the equation $E'(V_i)V_i=0$ and the Pohozaev identity one gets
\begin{equation}
\label{eulp}
\|V_i\|^{p+1}_{L^{p+1} |
}= 2\,\frac{p+1}{p-1}\,E(V_i),\qquad
\|V_i\|^2_{L^2}= \frac{N+2-p\,(N-2)}{p-1}\,E(V_i)
\end{equation}
Since the ground state solution $Z$ satisfies the same identities, the bounds \eqref{uppstiml2} are readily verified.
\end{proof}
\begin{proposition}
Let | \eqref{eq:mainass_secMorse} hold and the functions $V_i$ be defined as in Lemma
\ref{lemlim1}. We have, as $n\to +\infty$,
\begin{eqnarray}
\label{convl2}
{\mu_n}^{\frac{2}{p-1}}\,\lambda_n^{N/2-2/(p-1)}&\longrightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}| |
V_i|^2\,dx
\\
\label{convlp}
{\mu_n}^{\frac{p+1}{p-1}}\,\lambda_n^{N/2-(p+1)/(p-1)}\int_{\Omega} |u_n|^{p+1} \,dx&\longrightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|V_i|^{p+1}\,dx
\\
\label{convl2grad}
\alpha_n\,{\mu_n}^{\frac{2}{p-1}}\,\lambda_n^{N/2-(p+ | 1)/(p-1)}&\longrightarrow
\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx.
\end{eqnarray}
\end{proposition}
\begin{proof}
The limits \eqref{convl2} and \eqref{convlp} follow respectively by choosing $q=2$ and $q=p+1$ in \eqref{convlq} (recall that $\ |
|u_n\|_{L^{2}}=1$). Furthermore, from the equations
for $u_n$ and $V_k$, we have
\[
\alpha_n+\lambda_n=\mu_n\|u_n\|_{L^{p+1}}^{p+1},\qquad
\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx + \int_{{\mathbb{R}}^n}|V_i|^2\,dx = \int_{{\mathbb{R}}^n}|V_i|^{p+1}\,dx,
\] |
and also \eqref{convl2grad} follows.
\end{proof}
\begin{corollary}
\label{limmass}
With the same assumptions as above, we have that
\begin{enumerate}
\item if $1<p<1+\frac{4}{N}$, then $\mu_n\to +\infty$
\item if $p=1+\frac{4}{N}$, then $\mu_n\to \big |
(\sum_{i=1}^{k}\|V_i\|_{L^2}^2\big )^{2/N}\ge k^{2/N}
\|Z\|_{L^2}^{4/N}$
\item if $1+\frac{4}{N}<p<2^*-1$, then $\mu_n\to 0$.
\end{enumerate}
Furthermore
\begin{equation}
\label{limalphalam}
\frac{\alpha_n}{\lambda_n}\longrightarrow \frac{N(p-1)}{N+2- | p(N-2)}.
\end{equation}
\end{corollary}
\begin{proof}
The limits of $\mu_n$ follow by the previous proposition. To prove the lower bound in $2$, recall that either $V_i=Z$ or $V_i$ satisfies \eqref{uppstiml2}. Finally, taking the quotient between \eqref{co |
nvl2grad} and \eqref{convl2}, we have
$$
\frac{\alpha_n}{\lambda_n}\longrightarrow
\frac{\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}|\nabla V_i|^2\,dx}{\sum_{i=1}^{k}\int_{{\mathbb{R}}^n}| V_i|^2\,dx}
$$
On the other hand, for every $i=1,2,...,k$ it holds
$$
\|\na | bla V_i\|_{L^2}^2=\Bigg (\frac{\| V_i\|_{L^{p+1}}^{p+1}}{\|V_i\|_{L^2}^2} -1 \Bigg )\|V_i\|_{L^2}^2=
\frac{N(p-1)}{N+2-p(N-2)}\, \|V_i\|_{L^2}^2
$$
where the last equality follows by \eqref{eulp}. By inserting this into the above limit, we get \eqref{limal |
phalam}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:bbd_index}]
Let $(U_n,\lambda_n)$ solve \eqref{eq:main_prob_U}, with $\rho=\rho_n\to +\infty$
and $m(U_n)\leq k$. Changing variables as in \eqref{eq:main_prob_u}, we have that
$u_n=\rho_n^{-1/2} | U_n$ satisfies \eqref{eq:auxiliary_n} with $\mu_n = \rho_n^{(p-1)/2} \to
+\infty$. As a consequence, Lemma \ref{lemma:case_alpha_n_bounded} guarantees that
$\alpha_n\to+\infty$, and Corollary \ref{limmass} yields $p<1+4/N$.
On the other hand, by direct mi |
nimization of the energy one can show that, if $p<1+4/N$, for
every $\rho>0$ there exists a solution of \eqref{eq:main_prob_U} having Morse index one (see
also Section \ref{sec:1const}).
\end{proof}
\begin{remark}
\label{limGN}
Reasoning as above we can al | so show that
\begin{equation}
\label{newcnp1}
\frac{\int_{\Omega} |u_n|^{p+1} \,dx}{\alpha_n^{N(p-1)/4}}\longrightarrow
C_{N,p}\,\frac{\|Z\|_{L^2}^{p-1}}{\big (\sum_{i=1}^{k}\| V_i\|_{L^2}^2\big )^{(p-1)/2}}.
\end{equation}
\end{remark}
\section{Max-min |
principles with two constraints}\label{sec:2const}
In this section we deal with the maximization problem with two constraints introduced in \cite{MR3318740}, aiming at considering more general max-min classes of critical points.
Let ${\mathcal{M}}$ be def | ined in \eqref{Emu} and, for any fixed $\alpha>\lambda_1(\Omega)$, let $\mathcal{B}_\alpha$, $\mathcal{U}_\alpha$ be defined as in \eqref{eq:defBU}. We will look for critical points of the $\mathcal{C}^2$ functional
\[
f(u)=\int_{\Omega}|u|^{p+1},\quad\qua |
d\quad u\in {\mathcal{M}},
\]
constrained to $\mathcal{U}_\alpha$. To start with, we notice that the topological
properties of such set depend on $\alpha$.
\begin{lemma}\label{lemma:tilde_U_manifold}
Let $\alpha>\lambda_1(\Omega)$. Then the set
\[
{\mathca | l{U}}_{\alpha}\setminus\left\{ \varphi\in {\mathcal{U}}_\alpha :
-\Delta\varphi = \alpha\varphi\right\}
\]
is a smooth submanifold of $H^1_0(\Omega)$ of codimension 2. In particular, this property
holds true for ${\mathcal{U}}_\alpha$ itself, provided $\al |
pha\neq\lambda_k(\Omega)$, for every $k$.
\end{lemma}
\begin{proof}
Let us set $F(u)=(\int_\Omega u^2\,dx-1, \ \int_\Omega|\nabla u|^2\,dx)$. For every
$u\in{\mathcal{U}}_\alpha$, if the range of $F'(u)$ is ${\mathbb{R}}^2$ then ${\mathcal{U}}_\alpha$ is a | smooth manifold
at $u$. Since
\[
F'(u)[v]=2\left(\int_\Omega uv\,dx, \ \int_\Omega\nabla u\cdot\nabla v\,dx\right),
\qquad\text{for every }v\in H^1_0(\Omega),
\]
and $F'(u)[u]=2(1,\alpha)$, we have that $F'(u)$ is not surjective if and only if
\[
\int_\Om |
ega\nabla u\cdot\nabla v\,dx = \alpha \int_\Omega uv\,dx
\qquad\text{for every }v\in H^1_0(\Omega). \qedhere
\]
\end{proof}
\begin{remark}
If $\varphi$ belongs to the eigenspace corresponding to $\lambda_k(\Omega)$, then
$\varphi \in {\mathcal{U}}_{\lambda | _k(\Omega)}$. As a consequence ${\mathcal{U}}_{\lambda_k(\Omega)}$ may not be smooth
near $\varphi$. For instance, ${\mathcal{U}}_{\lambda_1(\Omega)}$ consists of two isolated points, $\pm\varphi_1$.
\end{remark}
Of course ${\mathcal{U}}_\alpha$ is closed |
and odd, for any $\alpha$. Recalling Definition
\ref{def:genus} we deduce that its genus
$\gamma({\mathcal{U}}_\alpha)$ is well defined.
\begin{lemma}
If $\alpha<\lambda_{k+1}(\Omega)$, for some $k$, then $\gamma({\mathcal{U}}_\alpha)\leq k$.
\end{lemma}
\ | begin{proof}
Let $V_k:=\spann\{\varphi_1,\dots,\varphi_k\}$.
Since
\[
\min\left\{\int_\Omega |\nabla u|^2\,dx : u\in V_k^\perp,\,
\int_\Omega u^2\,dx=1\right\}=\lambda_{k+1}(\Omega),
\]
we have that ${\mathcal{U}} \cap V_k^\perp = \emptyset$, thus the proj |
ection
\[
g := \proj_{V_k} \colon {\mathcal{U}}_\alpha \to V_k\setminus\{0\}
\]
is a continuous odd map of ${\mathcal{U}}_\alpha$ into $V_k\setminus\{0\}$. Now, let $h\colon{\mathbb{S}}^{m}\to {\mathcal{U}}$ be continuous and odd.
Then $g\circ h$ is contin | uous and odd from ${\mathbb{S}}^{m}$ to $V_k\setminus\{0\}$, and Borsuk-Ulam's Theorem forces $m\leq k-1$.
\end{proof}
\begin{lemma}\label{lemma:genusbigger}
If $\alpha>\lambda_{k}(\Omega)$, for some $k$, then $\gamma({\mathcal{U}}_\alpha)\geq k$.
\end{lem |
ma}
\begin{proof}
To prove the lemma we will construct a continuous map $h\colon {\mathbb{S}}^{k-1} \to {\mathcal{U}}$. Let
$\ell\in{\mathbb{N}}$ be such that $\lambda_{\ell+1}(\Omega)>\alpha$. For every $i=1,\dots,k$ we define the
functions
\[
u_i:=\left( | \frac{\lambda_{\ell+i}(\Omega)-\alpha}{\lambda_{\ell+i}(\Omega)-\lambda_i(\Omega)}\right)^{1/2}\varphi_i
+\left(\frac{\alpha-\lambda_{i}(\Omega)}{\lambda_{\ell+i}(\Omega)-\lambda_i(\Omega)}\right)^{1/2}\varphi_{\ell+i}.
\]
We obtain the following straightf |
orward consequences:
\begin{enumerate}
\item as $\lambda_i(\Omega)<\alpha<\lambda_{\ell+i}(\Omega)$, for every $i$, $u_i$ is well defined;
\item $\int_\Omega u_i^2\,dx=1$, $\int_\Omega |\nabla u_i|^2\,dx=\alpha$;
\item for every $j\neq i$ it holds $\int | _\Omega u_iu_j\,dx=\int_\Omega \nabla u_i\cdot\nabla u_j
\,dx=0$.
\end{enumerate}
Therefore the map $h\colon {\mathbb{S}}^{k-1} \to {\mathcal{U}}$ defined as
\[
h\colon x=(x_1,\dots,x_k) \mapsto \sum_{i=1}^k x_iu_i
\]
has the required properties.
\end{pro |
of}
Now we turn to the properties of the functional $f$. To start with, it satisfies
the Palais-Smale (P.S. for short) condition on $\overline{\mathcal{B}}_{\alpha}$; more precisely, the following
holds.
\begin{lemma}
\label{psball}
Every P.S. sequence $u | _n$ for $f\big |_{\overline{\mathcal{B}}_{\alpha}}$ is a P.S.
sequence for $f\big |_{\mathcal{U}_{\alpha}}$ and has a strongly convergent
subsequence in $\mathcal{U}_{\alpha}$.
\end{lemma}
\begin{proof}
We first show that there are no P.S. sequences in ${ |
\mathcal{B}_{\alpha}}$. In fact, if $u_n$ is such a sequence, there is a sequence of real numbers $k_n$ such that
\begin{equation}
\label{ps}
\int_{\Omega}|u_n|^{p-1}u_n\,v-k_n\int_{\Omega}u_n\,v=o(1)\,\|v\|_{H^1_0}
\end{equation}
for every $v\in H^1_0(\Om | ega)$. Since $u_n$ is bounded in $H^1_0(\Omega)$, there is a subsequence (still denoted by $u_n$)
weakly convergent to $u\in H^1_0(\Omega)$; moreover, $u_n$ converges strongly in $L^{p+1}(\Omega)$ and in $L^2(\Omega)$ to the same limit. By choosing $v=u_n$ |
, we see that $k_n$ is bounded, so that we can also assume that $k_n\rightarrow k$. By taking the limit of \eqref{ps} for $n\to\infty$ we get
\begin{equation}
\nonumber
\int_{\Omega}|u|^{p-1}u\,v=k\int_{\Omega}u\,v
\end{equation}
for every $v\in H^1_0(\Ome | ga)$. Hence $u$ is constant, but this contradicts $u\in {\mathcal{M}}$.
Now, if $u_n$ is a P.S. sequence for $f$ on ${\mathcal{U}}_{\alpha}$, there are sequences of real numbers $k_n$, $l_n$ such that
\begin{equation}
\label{ps1}
\int_{\Omega}|u_n|^{p-1}u |
_n\,v-k_n\int_{\Omega}u_n\,v-l_n\int_{\Omega}\nabla u_n\,\nabla v=o(1)\,\|v\|_{H^1_0}.
\end{equation}
It is readily seen that $l_n$ is bounded away from zero, otherwise \eqref{ps1} is equivalent to \eqref{ps} (for some subsequence) and we still reach a con | tradiction. Then, we can divide both sides by $l_n$ and find that
there are sequences $\{\lambda_n\}_n$, $\{\mu_n\}_n$, with $\mu_n$ bounded, such that
\begin{equation}
\nonumber
\int_{\Omega}\nabla u_n\,\nabla v+\lambda_n\int_{\Omega}u_n\,v-\mu_n\int_{\Om |
ega}|u_n|^{p-1}u_n\,v=o(1)\,\|v\|_{H^1_0}.
\end{equation}
Now, by reasoning as before one finds that
also the sequence $\{\lambda_n\}_n$ is bounded, so that by the relation
$$-\Delta u_n+\lambda_n u_n-\mu_n |u_n|^{p-1}u_n=o(1)\quad \mathrm{in}\,\, H^{-1}(\ | Omega)$$
and by the compactness of the embedding $H^1_0(\Omega)\hookrightarrow L^{p+1}(\Omega)$, the P.S. condition holds for the functional $f\big |_{{\mathcal{U}}_{\alpha}}$.
\end{proof}
We can combine the previous lemmas to prove one of the main result |
s stated in the introduction.
\begin{proof}[Proof of Theorem \ref{thm:genus_2constr}]
Lemma \ref{psball} allows to apply standard variational methods (see e.g.
\cite[Thm. II.5.7]{St_2008}). We deduce that
$M_{\alpha,\,k}$ is achieved
at some critical point | $u$ of $f\big |_{\mathcal{U}_{\alpha}}$.
This amounts to say that $u$ satisfies \eqref{lagreq}
for some real $\lambda$ and $\mu\neq 0$. We claim that there exists at least one
$u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$ such that \eqref{lagreq} |
holds with $\mu>0$.
Assume by contradiction that for \emph{every} critical point of $f\big |_{\mathcal{U}_{\alpha}}$ at level $M_{\alpha,k}$ it holds $\mu< 0$ in equation \eqref{lagreq}.
Let us define the functional $T:\,H^1_0(\Omega)\to {\mathbb{R}}$ as |
$$T(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2.$$
By denoting with $D$ the Fr\'{e}chet derivative and by $<\,,\,>$ the pairing between $H_0^1$ and its dual $H^{-1}$, our assumption can be restated as follows:
\noindent if there are $u\in f^{-1}(M_{\alpha,\, |
k})\cap\mathcal{U}_{\alpha}$ and $\mu\neq 0$ such that
\begin{equation}
\label{lagreq1}
\langle DT(u),\phi\rangle=\mu\langle Df(u),\phi\rangle
\end{equation}
for every $\phi\in H^1_0(\Omega)$ satisfying $\int_{\Omega}\phi u=0$ (that is for every $\phi$ ta | ngent
to ${\mathcal{M}}$ at $u$) then $\mu<0$.
We stress that both $DT(u)$ and $Df(u)$ in the above equation are bounded away from zero, since there are no Dirichlet eigenfunctions in $\mathcal{U}_{\alpha}$ nor critical points of $f$ on ${\mathcal{M}}$.
H |
ence, by denoting with $\nabla_{T{\mathcal{M}}}$ the gradient of a functional (in $H^1_0$) in the direction tangent to ${\mathcal{M}}$, if $u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$ then $\nabla_{T{\mathcal{M}}}T(u)$ and
$\nabla_{T{\mathcal{M}} | }f(u)$ \emph{are either opposite or not parallel}. Moreover, the angle between these (non vanishing) vectors is \emph{bounded away from zero}; otherwise, we would find sequences $u_n\in \mathcal{U}_{\alpha}$, $\mu_n>0$ such that
\begin{equation}
\label{no |
paral}
(\nabla_{T{\mathcal{M}}}T(u_n),v)_{H^1_0}-\mu_n(\nabla_{T{\mathcal{M}}}f(u_n),v)_{H^1_0}=o(1)\|v\|_{H^1_0}
\end{equation}
for every $v\in H^1_0(\Omega)$; but since
$$(\nabla_{T{\mathcal{M}}}T(u_n),v)_{H^1_0}=\int_{\Omega}\nabla u_n\,\nabla v-\lambda | _n^T\int_{\Omega}u_n\, v\,,$$
$$(\nabla_{T{\mathcal{M}}}f(u_n),v)_{H^1_0}=\int_{\Omega}|u_n|^{p-1}u_n\,v-\lambda_n^f\int_{\Omega}u_n\, v\,,$$
for suitable bounded sequences $\lambda_n^T$, $\lambda_n^f$, this is equivalent to saying that $u_n$ is a P.S. seq |
uence for
$f\big |_{\mathcal{U}_{\alpha}}$, so that, by Lemma \ref{psball}, we would get a constrained critical point with $\mu>0$.
Then, by choosing suitable linear combinations of the above tangential components
one can define a bounded $\mathcal{C}^1$ | map $u\mapsto v(u)\in H_0^1(\Omega)$, with $v(u)$ tangent to ${\mathcal{M}}$
and satisfying the following property: there is $\delta>0$ such that
\begin{equation}
\label{diseqv}
\int_{\Omega}\nabla u\,\nabla v(u)< -\delta\, ,\quad\quad \int_{\Omega}|u|^{p- |
1}u\,v(u)>\delta\,,
\end{equation}
for every $u\in f^{-1}(M_{\alpha,\,k})\cap \mathcal{U}_{\alpha}$. By continuity and possibly by decreasing $\delta$, inequalities \eqref{diseqv} extend to
\begin{equation}
\label{diseqv1}
f^{-1}(M_{\alpha,\,k}-\bar\vareps | ilon, M_{\alpha,\,k}+\bar\varepsilon)\cap \big (\overline{\mathcal{B}}_{\alpha}
\backslash \overline{\mathcal{B}}_{\alpha-\tau}\big )
\end{equation}
for small enough, positive $\bar\varepsilon$ and $\tau$. Finally, since there are no critical points of $f$ |
in
${\mathcal{B}}_{\alpha}$ we can take that the \emph{second of \eqref{diseqv} holds on}
\begin{equation}
\label{diseqv2}
f^{-1}(M_{\alpha,\,k}- \bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon)\cap \overline{\mathcal{B}}_{\alpha}.
\end{equation}
Let $\v | arphi$ be a $\mathcal{C}^1$ function on ${\mathbb{R}}$ such that:
$$0\le\varphi\le 1, \quad\varphi\equiv 1\,\, \mathrm{in}\,\,(M_{\alpha,\,k}-\bar\varepsilon/2, M_{\alpha,\,k}+\bar\varepsilon/2),\quad \varphi\equiv 0\,\, \mathrm{in}\,\,{\mathbb{R}}\backsla |
sh (M_{\alpha,\,k}-\bar\varepsilon, M_{\alpha,\,k}+\bar\varepsilon),$$
and define
\begin{equation}
\label{vectfield}
e(u)=\varphi(f(u))\,v(u).
\end{equation}
Clearly, $e$ is a $\mathcal{C}^1$ vector field on ${\mathcal{M}}$ and is uniformly bounded, so th | at there exists a global solution $\Phi(u,t)$ of the initial value problem
$$\partial_t\Phi(u,t)=e\big (\Phi(u,t)),\quad\quad \Phi(u,0)=0.$$
By definition \eqref{vectfield} and by the first of \eqref{diseqv} (on \eqref{diseqv1}) we get
$\Phi(u,t_0) \in \o |
verline{\mathcal{B}}_{\alpha}$ for $t_0> 0$ and for any $u\in \overline{\mathcal{B}}_{\alpha}$; moreover, by the second inequality of \eqref{diseqv} (on \eqref{diseqv2}) there exists $\varepsilon\in (0,\bar\varepsilon)$ such that
$$f(\Phi(u,t_0))>M_{\alpha | ,\,k}+\varepsilon$$ for every $u\in f^{-1}(M_{\alpha,\,k}-\varepsilon, +\infty)\cap \overline{\mathcal{B}}_{\alpha}$.
Now, by \eqref{maxmin}, there is $A_{\varepsilon}\subset \overline{\mathcal{B}}_{\alpha}$ such that $\gamma(A_{\varepsilon})\ge k$ and
$$ |
\inf_{u\in A_{\varepsilon}} f(u)\ge M_{\alpha,\,k}-\varepsilon.$$
Hence, $\gamma\big (\Phi(A_{\varepsilon},t_0) \big )\ge k$ and
$$\inf_{u\in \Phi(A_{\varepsilon},t_0)} f(u)\ge M_{\alpha,\,k}+\varepsilon$$
contradicting the definition of $M_{\alpha,\,k}$.
| \end{proof}
\begin{remark}
\label{rem1}
If $\mu>0$, by testing \eqref{lagreq} with $u$ and by integration by parts we readily get
$\lambda>-\alpha$. An alternative lower bound, independent of $\alpha$, could be obtained
by adapting arguments from \cite{MR9 |
68487,MR954951,MR991264} in order to prove
that the Morse index of $u$ (as a solution of \eqref{lagreq}) is less or equal than $k$.
Then Lemma \ref{lem:lambda_bdd_below} would provide $\lambda\geq-\lambda_{k}$.
\end{remark}
\begin{remark}\label{rem:MvsCNp} |
By the Gagliardo-Nirenberg inequality \eqref{sobest} we readily obtain that, for every $k\geq1$,
\[
M_{\alpha,k}\leq C_{N,p} \alpha^{N(p-1)/4}.
\]
Taking into account the previous remark, this agrees with Remark \ref{limGN}.
\end{remark}
We conclude this |
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