\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 135, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2016/135\hfil Multiple sign-changing solutions]
{Multiple sign-changing solutions for Kirchhoff type problems}
\author[C. J. Batkam \hfil EJDE-2016/135\hfilneg]
{Cyril Joel Batkam}
\address{Cyril Joel Batkam \newline
HEC Montreal,
3000 Chemin de la Cate-Sainte-Catherine,
Montrel, QC, H3T 2B1, Canada}
\email{cyril-joel.batkam@hec.ca, cyril.joel.batkam@usherbrooke.ca}
\thanks{Submitted December 17, 2015. Published June 7, 2016.}
\subjclass[2010]{35J60, 35A15, 35J20, 35J25}
\keywords{Kirchhoff type equation; sign-changing solution;
multiple solutions; \hfill\break\indent critical point theorem}
\begin{abstract}
This article concerns the existence of sign-changing solutions
to nonlocal Kirchhoff type problems of the form
\[
-\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \text{ in }\Omega,\quad
u=0 \text{ on }\partial\Omega,
\]
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$)
with smooth boundary, $a>0$, $b\geq0$, and
$f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a continuous function.
We first establish a new sign-changing version of the symmetric mountain
pass theorem and then apply it to prove the existence of a sequence of
sign-changing solutions with higher and higher energy.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
In this article, we study the multiplicity of sign-changing solutions to
nonlocal Kirchhoff type problems of the form
\begin{equation}\label{s}
\begin{gathered}
-\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\quad \text{in }\Omega,\\
u=0 \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$) with
smooth boundary, $a>0$, $b\geq0$, and
$f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a nonlinear function.
We restrict $N\leq3$ because $f(x,u)$ will behave as
$|u|^p$ with $4\leq p<2^\star$, where $2^\star=2N/(N-2)$ is the critical
Sobolev exponent. This will allow us to attack the problem using
variational methods.
Problem \eqref{s} is related to the stationary analogue of the hyperbolic equation
\[
u_{tt}-\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u),
\]
which is a general version of the equation
\begin{equation}\label{kir}
\rho\frac{\partial^2u}{\partial t^2}
-\Big(\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L\big|
\frac{\partial u}{\partial x}\big|^2dx\Big)\frac{\partial^2u}{\partial x^2}=0
\end{equation}
proposed by Kirchhoff \cite{Kir} as an extension of the classical D'Alembert's
wave equation for free vibrations of elastic strings. This model takes into
account the changing in length of the string produced by transverse vibrations.
In \eqref{kir}, $L$ is the length of the string, $h$ is the area of the
cross-section, $E$ is the Young's modulus of the material, $\rho$ is
the mass density, and $\rho_0$ is the initial tension.
When $b>0$, problem \eqref{s} is said to be nonlocal.
In that case, the first equation in \eqref{s} is no longer a pointwise equality.
This causes some mathematical difficulties which make the study of such problems
particularly interesting. Some early classical studies of Kirchhoff type
problems can be found in \cite{Bern,Poho}. However, problem \eqref{s} received
much attention only after the paper of Lions \cite{Lions}, where an abstract
framework to attack it was introduced. Some existence and multiplicity results
can be found in \cite{B13,He-Zou,LiuHe12,Per-Zhang} without any information
on the sign of the solutions. Recently, Alves et al \cite{Alves05},
Ma and Rivera \cite{Ma-Riv03}, and Cheng and Wu \cite{Cheng-Wu} obtained
one positive solution. In \cite{HeZou09}, He and Zou obtained infinitely
many positive solutions. The existence of sign-changing solutions to
\eqref{s} was considered by Figuereido and Nascimento \cite{Figue},
Perera and Zhang \cite{Per-Zhang06}, Mao and Zhang \cite{Mao-Zhang09},
and Mao and Luan \cite{Mao-Luan11}. But only one sign-changing solution
was found in these papers. In case $f$ is a pure power type nonlinearity,
Alves et al \cite{Alves05} related the number of solutions of \eqref{s}
to that of a local problem by using a scaling argument.
As a consequence, one can obtain in that particular case infinitely many
sign-changing solutions (see \cite{WHL}). However, the scaling approach
does not provide high energy solutions even in the simple case of power
type nonlinearity.
In this article, we develop a variational approach to study high-energy
sign-changing solutions to some classes of nonlocal problems.
Our result on \eqref{s} relies on the following standard conditions
on the nonlinear term $f$:
\begin{itemize}
\item[(H1)] $f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$
is continuous and there exists a constant $c>0$ such that
\[
|f(x,u)|\leq c\big(1+|u|^{p-1}\big),
\]
where $p>4$ for $N=1,2$ and $4
4$ such that $0<\mu F(x,u)\leq uf(x,u)$
for all $u\neq0$ and for a.e $x\in\overline{\Omega}$, where
$F(x,u)=\int_0^u f(x,s)ds$.
\item [(H4)] $f(x,-u)=-f(x,u)$ for all $(x,u)\in\overline{\Omega}\times\mathbb{R}$.
\end{itemize}
One can verify easily that the function $f(x,u)=|u|^p$,
with $p$ as in condition (H1), satisfies the above conditions.
Our result reads as follows:
\begin{theorem}\label{mainresult}
Let $a>0$ and $b\geq0$. Assume that $f$ satisfies the conditions
{\rm(H1)--(H4)}.
Then \eqref{s} possesses a sequence $(u_k)$ of sign-changing solutions such that
\[
\frac{a}{2}\int_\Omega|\nabla u_k|^2dx+\frac{b}{4}
\Big(\int_\Omega|\nabla u_k|^2dx\Big)^2
-\int_\Omega F(x,u_k)dx\to+\infty,\quad \text{as }k\to\infty.
\]
\end{theorem}
If $b=0$, we obtain the following consequence of the above result.
\begin{corollary}\label{zero}
Under assumptions {\rm (H1)--(H4)}, the semilinear problem
\begin{equation}\label{classic}
\begin{gathered}
-\Delta u=f(x,u)\quad \text{ in }\Omega,\\
u=0 \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
possesses a sequence $(u_k)$ of sign-changing solutions such that
\[
\frac{1}{2}\int_\Omega|\nabla u_k|^2dx-\int_\Omega F(x,u_k)dx\to+\infty,\quad \text{as }k\to\infty.
\]
\end{corollary}
Note that Corollary \ref{zero} was obtained by Qian and Li \cite{QianLi04}
by means of the method of invariant sets of descending flow.
Earlier proofs were also given in \cite{Bar01,LW02} under the stronger
assumption that $f$ is smooth. The arguments of \cite{QianLi04,Bar01,LW02}
rely on sign-changing critical point theorems built only for functionals
of the form
\[
u\in H_0^1(\Omega)\mapsto \frac{1}{2}\|u\|^2-\Psi(u),
\]
where $\Psi'$ is completely continuous, and cannot then be applied to
\eqref{s} when $b>0$. Hence our result in Theorem \ref{mainresult} can
be regarded as an extension of the classical result for the semilinear
problem \eqref{classic} to the case of the nonlinear Kirchhoff type
problem \eqref{s}. We also mention here that the result of Theorem \ref{mainresult}
was more or less expected. However, it seems that this paper is the first
to provide a formal proof. Moreover, we believe that the critical point theorem
we will establish in the next section is of independant interest and can be
applied to many other nonlocal problems (indeed, some applications by the
author and collaborators will appear in other journals).
The study of sign-changing solutions is related to several long-standing
questions concerning the multiplicity of solutions for elliptic boundary value
problems. Compared with positive and negative solutions, sign-changing
solutions have more complicated qualitative properties and are more
difficult to find. During the last thirty years, several sophisticated
techniques in calculus of variations and in critical point theory
were developed to study the multiplicity of sign-changing solutions to
nonlinear elliptic partial differential equations.
In \cite{Bar01} and \cite{LW02}, the authors established some multiplicity
sign-changing critical point theorems in partially ordered Hilbert spaces
by using Morse theory and the method of invariant sets of descending flow
respectively. In \cite{Zou06}, a parameter-depending sign-changing fountain
theorem was established without any Palais-Smale type assumption.
More recently, a symmetric mountain pass theorem in the presence of invariant
sets of the gradient flows was introduced in \cite{LLW}.
However, it seems that all these powerful approaches are not directly applicable
to find multiple sign-changing solutions to \eqref{s}.
Our approach in proving Theorem \ref{mainresult} relies on a new sign-changing
critical point theorem, also established in this paper, which is modelled on
the fountain theorem of Bartsch (see \cite[Theorem 3.6]{W}).
An essential tool in the proof of this theorem is a deformation lemma,
which allows to lower sub-level sets of a functional, away from its critical set.
The main ingredient in the proof of the deformation lemma is a suitable negative
pseudo-gradient flow, a notion introduced by Palais \cite{Palais}.
Since we are interesting in sign-changing critical points, the pseudo-gradient
flow must be constructed in such the way that it keeps the positive and negative
cones invariant. This invariance property makes the construction of the flow very
complicated when the problem contains nonlocal terms. In this paper, we borrow
some ideas from recent work by Liu, Liu and Wang \cite{LLW} on the nonlinear
Sch\"{o}dinger systems and by Liu, Wang and Zhang \cite{LZW} on the
nonlinear Sch\"{o}dinger-Poisson system, where the pseudo-gradient flows were
constructed by using an auxiliary operator. However, the critical point theorem
used in \cite{LLW,LZW} cannot be applied to prove Theorem \ref{mainresult}
because the corresponding auxiliary operator in the case of \eqref{s} is not compact.
The rest of this article is organized as follows. In Section \ref{deux},
we state and prove the new sign-changing critical point theorem.
In Section \ref{trois}, we provide the proof of Theorem \ref{mainresult}.
Throughout this article, we denote by ``$\to$'' the strong converge and by
``$\rightharpoonup$'' the weak convergence.
\section{An abstract sign-changing critical point theorem for even functionals}
\label{deux}
In this section, we present a variant of the symmetric mountain pass type
theorem which produces a sequence of sign-changing critical points with
arbitrary large energy.
Let $\Phi$ be a $C^1$-functional defined on a Hilbert space $X$ of the form
\begin{equation}\label{space}
X:=\overline{\oplus_{j=0}^\infty X_j},\quad\text{with } \dim X_j<\infty.
\end{equation}
We introduce for $k\geq2$ and $m>k+2$ the following notation:
\begin{gather*}
Y_k:=\oplus_{j=0}^k X_j,\quad
Z_k=\overline{\oplus_{j=k}^\infty X_j},
\quad Z^m_k=\oplus_{j=k}^m X_j,\quad
B_k:=\big\{u\in Y_k: \|u\|\leq\rho_k\big\},
\\
N_k:=\big\{u\in Z_k:\|u\|=r_k\big\},\quad
N^m_k:=\big\{u\in Z^m_k:\|u\|=r_k\big\},\text{ where }00$ we set
\[
\pm D_m^0:=\big\{u\in Y_m: \operatorname{dist}\big(u,\pm P_m\big)<\mu_m\big\},\quad
D_m=D_m^0\cup(-D_m^0),\quad
S_m:=Y_m\backslash D_m.
\]
We will also denote the $\alpha$-neighborhood of $S\subset Y_m$ by
\[
V_\alpha(S):=\big\{u\in Y_m\,|\operatorname{dist}(u,S)\leq\alpha\big\},\quad
\forall\alpha>0.
\]
Let us now state our critical point theorem. It is a version of the symmetric
mountain pass theorem of Ambrosetti and Rabinowitz \cite{A-R}, and we model
it on the fountain theorem of Bartsch \cite{B}.
\begin{theorem}[Sign-changing fountain theorem]\label{scft}
Let $\Phi\in C^1(X,\mathbb{R})$ be an even functional which maps
bounded sets to bounded sets. If, for $k\geq2$ and $m>k+2$, there exist
$00$ such that
\begin{itemize}
\item[(H5)] $a_k:=\max_{u\in \partial B_k} \Phi(u)\leq0$ and
$b_k:=\inf_{u\in N_k} \Phi(u)\to+\infty$, as $k\to\infty$.
\item[(H6)] $N^m_k\subset S_m$.
\item[(H7)] There exists an odd locally Lipschitz continuous vector field
$B:E_m\to Y_m$ such that:
\begin{itemize}
\item[(i)] $B\big((\pm D_m^0)\cap E_m\big)\subset \pm D_m^0$;
\item[(ii)] there exists a constant $\alpha_1>0$ such that
$\langle \Phi'_m(u),u-B(u)\rangle \geq\alpha_1\|u-B(u)\|^2$, for any $u\in E_m$;
\item[(iii)] for $a0$, there exists $\beta>0$ such that
$\|u-B(u)\|\geq\beta$ if $u\in Y_m$ is such that $\Phi_m(u)\in[a,b]$
and $\|\Phi'_m(u)\|\geq\alpha$.
\end{itemize}
\item[(H8)] $\Phi$ satisfies the $(PS)^\star_{nod}$ condition, that is:
\begin{itemize}
\item[(i)] any Palais-Smale sequence of $\Phi_m$ is bounded;
\item[(ii)] any sequence $(u_{m_j})\subset X$ such that
$m_j\to\infty$, $u_{m_j}\in V_{\mu_{m_j}}(S_{m_j})$,
$\sup\Phi(u_{m_j})<\infty$, and $\Phi'_{m_j}(u_{m_j})=0$,
has a subsequence converging to a sign-changing critical point of $\Phi$.
\end{itemize}
\end{itemize}
Then $\Phi$ has a sequence $(u_k)_k$ of sign-changing critical points in
$X$ such that $\Phi(u_k)\to\infty$, as $k\to\infty$.
\end{theorem}
Condition (H8) is a version of the usual compactness
condition in critical point theory, namely the Palais-Smale condition.
We recall that a sequence $(u_n)\subset E$ is a Palais-Smale sequence of
a smooth functional $J$ defined on a Banach space $E$ if the sequence
\big($J(u_n)\big)$ is bounded and $J'(u_n)\to0$, as $n\to\infty$.
If every such sequence possesses a convergent subsequence, then $J$
is said to satisfy the Palais-Smale condition.
We need a special deformation lemma to prove the above result.
We first recall the following helpful lemma.
\begin{lemma}[{\cite[Lemma 2.2]{Zou06}}] \label{helpful}
Let $\mathcal{M}$ be a closed convex subset of a Banach space $E$.
If $H:\mathcal{M}\to E$ is a locally Lipschitz continuous map such that
\[
\lim_{\beta\to0^+} \frac{\operatorname{dist}\big(u+\beta H(u),\mathcal{M}\big)}
{\beta}=0,\quad\forall u\in\mathcal{M},
\]
then for any $u_0\in\mathcal{M}$, there exists $\delta>0$ such that the
initial value problem
\[
\frac{d\sigma(t,u_0)}{dt}=H\big(\sigma(t,u_0)\big),\quad \sigma(0,u)=u_0,
\]
has a unique solution defined on $[0,\delta)$. Moreover,
$\sigma(t,u_0)\in\mathcal{M}$ for all $t\in[0,\delta)$.
\end{lemma}
Now we state a quantitative deformation lemma.
\begin{lemma}[Deformation lemma]\label{defor}
Let $\Phi\in C^1(X,\mathbb{R})$ be an even functional which maps bounded
sets to bounded sets. Fix $m$ sufficiently large and assume that the condition
{\rm (H7)} holds. Let $c\in\mathbb{R}$ and $\varepsilon_0>0$ such that
\begin{equation}\label{one}
\forall u\in\Phi_m^{-1}\big([c-2\varepsilon_0,c+2\varepsilon_0]\big)
\cap V_{\frac{\mu_m}{2}}(S_m)\,:\, \|\Phi_m'(u)\|\geq\varepsilon_0.
\end{equation}
Then for some $\varepsilon\in]0,\varepsilon_0[$ there exists
$\eta\in C\big([0,1]\times Y_m,Y_m\big)$ such that
\begin{itemize}
\item[(i)] $\eta(t,u)=u$ for $t=0$ or
$u\notin \Phi_m^{-1}\big([c-2\varepsilon,c+2\varepsilon]\big)$;
\item[(ii)] $\eta\big(1,\Phi_m^{-1}(]-\infty,c+\varepsilon])
\cap S_m\big)\subset \Phi_m^{-1}\big(]-\infty,c-\varepsilon]\big)$;
\item[(iii)] $\Phi_m\big(\eta(\cdot,u)\big)$ is not increasing, for any $u$;
\item[(iv)] $\eta([0,1]\times D_m)\subset D_m$;
\item[(v)] $\eta(t,\cdot)$ is odd, for any $t\in[0,1]$.
\end{itemize}
\end{lemma}
\begin{proof}
Define $V:E_m\to Y_m$ by $V(u)=u-B(u)$, where $B$ is given by (H7).
Then there is $\delta>0$ such that $V(u)\geq\delta$ for any
$u\in\Phi_m^{-1}\big([c-2\varepsilon_0,c+2\varepsilon_0]\big)
\cap V_{\frac{\mu_m}{2}}(S_m)$ \big(in view (H7)-(iii)\big).
We take $\varepsilon\in ]0,\min(\varepsilon_0,\frac{\delta\alpha_1\mu_m}{8})[$
and we define
\begin{gather*}
A_1:=\Phi_m^{-1}\big([c-2\varepsilon,c+2\varepsilon]\big)
\cap V_{\frac{\mu_m}{2}}(S_m),\quad
A_2:=\Phi_m^{-1}\big([c-\varepsilon,c+\varepsilon]\big)\cap V_{\frac{\mu_m}{4}}(S_m),
\\
\chi(u):=\frac{\operatorname{dist}(u,Y_m\backslash A_1)}
{\operatorname{dist}(u,Y_m\backslash A_1)+\operatorname{dist}(u,A_2)}, \quad
u\in Y_m
\end{gather*}
so that $\chi=0$ on $Y_m\backslash A_1$, $\chi=1$ on $A_2$, and $0\leq\chi\leq1$.
We consider the vector field
\[
W(u):= \begin{cases}
\chi(u)\|V(u)\|^{-2}V(u), &\text{for }u\in A_1 \\
0, &\text{for }u\in Y_m\backslash A_1.
\end{cases}
\]
Clearly $W$ is odd and locally Lipschitz continuous. Moreover, by our choice
of $\varepsilon$ above we have
\begin{equation}\label{fourr}
\|W(u)\|\leq\frac{1}{\delta}\leq\frac{\alpha_1\mu_m}{8\varepsilon},\quad
\forall u\in Y_m.
\end{equation}
It follows that the Cauchy problem
\[
\frac{d}{dt}\sigma(t,u)=-W(\sigma(t,u)),\quad\sigma(0,u)=u\in Y_m
\]
has a unique solution $\sigma(\cdot, u)$ defined on $\mathbb{R}_+$.
Moreover, $\sigma$ is continuous on $\mathbb{R}_+\times Y_m$ and the map
$\sigma(t,\cdot):Y_m\to Y_m$ is a homeomorphism for each $t\geq0$
(see, for instance \cite{W}).
In view of \eqref{fourr}, we have
\begin{equation}\label{two}
\|\sigma(t,u)-u\|\leq \int_0^t\|W(\sigma(s,u))\|ds
\leq\frac{\alpha_1\mu_m}{8\varepsilon}t,
\end{equation}
and by (H7)-(ii)
\begin{equation}
\begin{aligned}
\frac{d}{dt}\Phi_m(\sigma(t,u))&=-\langle \Phi_m'(\sigma(t,u)),
\chi(\sigma(t,u))\|V(\sigma(t,u))\|^{-2}V(\sigma(t,u))\rangle \\
& \leq -\alpha_1\chi(\sigma(t,u)).
\end{aligned} \label{three}
\end{equation}
Define
\[
\eta:[0,1]\times Y_m\to Y_m,\quad
\eta(t,u):=\sigma\big(\frac{2\varepsilon}{\alpha_1}t,u\big).
\]
Conclusion (i) of the lemma is clearly satisfied and by \eqref{three}
above (iii) is also satisfied. Since $W$ is odd, (v) is a consequence of
the uniqueness of the solution to the above Cauchy problem.
We now verify (ii). Let $v\in\eta(1,\Phi_m^{-1}(]-\infty,c+\varepsilon])\cap S_m)$.
Then $v=\eta(1,u)=\sigma(\frac{2\varepsilon}{\alpha_1},u)$, where
$u\in \Phi_m^{-1}(]-\infty,c+\varepsilon])\cap S_m$.
If there exists $t\in[0,\frac{2\varepsilon}{\alpha_1}]$ such that
$\Phi_m(\sigma(t,u))0$ such that $\sigma(t_0,u_0)\notin D_m^0$.
Choose a neighborhood $N_{u_0}$ of $u_0$ such that $N_{u_0}\subset D_m^0$.
Then there is a neighborhood $N_0$ of $\sigma(t_0,u_0)$ such that
$\sigma(t_0,\cdot):N_{u_0}\to N_0$ is a homeomorphism
(because $\sigma(t_0,\cdot):Y_m\to Y_m$ is a homeomorphism).
Since $\sigma(t_0,u_0)\notin D_m^0$, the set $N_0\backslash \overline{D_m^0}$
is not empty. Hence there is $w\in N_{u_0}$ such that
$\sigma(t_0,w)\in N_0\backslash \overline{D_m^0}$, contradicting \eqref{eight}.
We now terminate by giving the proof of our above claim.
By (H7)-(i) we have $B( D_m^0\cap E_m)\subset D_m^0$, which implies
that $B( \overline{D_m^0}\cap E_m)\subset \overline{D_m^0}$.
Since $K_m\cap A_1=\emptyset$, we have $\sigma(t,u)=u$ for all
$t\in[0,1]$ and $u\in \overline{D^0_m}\cap K_m$.
Assume that $u\in \overline{D^0_m}\cap E_m$. If there is $t_1\in(0,1]$
such that $\sigma(t_1,u)\notin \overline{D_m^0}$, then there would be
$s_1\in[0,t_1)$ such that $\sigma(s_1,u)\in\partial\overline{D_m^0}$ and
$\sigma(t,u)\notin \overline{D_m^0}$ for all $t\in(s_1,t_1]$.
The Cauchy problem
\[
\frac{d}{dt}\mu(t,\sigma(s_1,u))=-W\big(\mu(t,\sigma(s_1,u))\big),\quad
\mu(0,\sigma(s_1,u))=\sigma(s_1,u)\in Y_m
\]
has $\sigma(t,\sigma(s_1,u))$ as unique solution. Recalling that $W=0$ on
$Y_m\backslash A_1$, we have $v-W(v)\in \overline{D_m^0}\cap (Y_m\backslash A_1)$
for any $v\in\overline{D_m^0}\cap(Y_m\backslash A_1)$.
Assume that $v\in A_1\cap \overline{D_m^0}$. Since $\|V(u)\|\geq\delta$,
we deduce that $1-\beta\chi(v)\|V(v)\|^{-2}\geq0$ for all $\beta$ such that
$0<\beta\leq \delta^2$. Recalling that $v\in\overline{D_m^0}$ implies
$\operatorname{dist}(v,P_m)\leq\mu_m$, that $V(v)=v-B(v)$, and that
$aP_m+bP_m\subset P_m$ for all $a,b\geq0$ \big(because $P_m$ is a cone\big),
we obtain for any $\beta\in]0,\delta^2]$
\begin{gather*}
\begin{aligned}
&\operatorname{dist}\big(v-\beta W(v),P_m\big)\\
&=\operatorname{dist}\big(v-\beta\chi(v)\|V(v)\|^{-2}V(v),P_m\big)\\
&=\operatorname{dist}\big(v-\beta\chi\|V(v)\|^{-2}(v-B(v)),P_m\big)\\
&=\operatorname{dist}\Big((1-\beta\chi(v)\|V(v)\|^{-2})v+\beta\chi(v)\|V(v)\|^{-2}B(v),P_m\Big)\\
&\leq \operatorname{dist}
\Big((1-\beta\chi(v)\|V(v)\|^{-2})v+\beta\chi(v)\|V(v)\|^{-2}B(v),\\
&\qquad \beta\chi(u)\|V(v)\|^{-2}P_m+(1-\beta\chi(v)\|V(v)\|^{-2})P_m\Big)\\
&\leq (1-\beta\chi(v)\|V(v)\|^{-2})\operatorname{dist}(v,P_m)
+\beta\chi(v)\|V(v)\|^{-2}\operatorname{dist}(B(v),P_m)\\
&\leq (1-\beta\chi(v)\|V(v)\|^{-2})\mu_m+\beta\chi(v)\|V(v)\|^{-2}\mu_m
=\mu_m.
\end{aligned}
\end{gather*}
It follows that $v-\beta W(v)\in\overline{D_m^0}$ for $0<\beta\leq \delta^2$.
This implies that
\[
\lim_{\beta\to0^+} \frac{\operatorname{dist}\big(v+\beta(- W(v)),
\overline{D_m^0}\big)}{\beta}=0,\quad\forall u\in\overline{D_m^0}.
\]
By Lemma \ref{helpful} there exists $\delta_0>0$ such that
$\sigma\big(t,\sigma(s_1,u)\big)\in \overline{D_m^0}$ for all
$t\in[0,\delta_0)$. This implies that
$\sigma\big(t,\sigma(s_1,u)\big)=\sigma(t+s_1,u)\in \overline{D_m^0}$
for all $t\in[0,\delta_0)$, which contradicts the definition of $s_1$.
This last contradiction assures that
$\sigma([0,+\infty)\times\overline{D_m^0})\subset\overline{D_m^0}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{scft}]
(H5) and (H6) imply that $a_k0$ such that
\[
\inf_{u\in Z_k\\\|u\|=r_k} \Phi(u)\to\infty,\text{ as }k\to\infty.
\]
\end{itemize}
\end{lemma}
\begin{proof}
(1) It is well known that integrating (H3) yields the existence of two
constants $c_1,c_2>0$ such that $F(x,u)\geq c_1|u|^\mu-c_2$.
This together with the fact that all norms are equivalent in the
finite-dimensional subspace $Y_k$ imply that
\[
\Phi(u)\leq \frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-c_3\|u\|^\mu
+c_4,\quad\forall u\in Y_k,
\]
where $c_3,c_4$ are positive constant. Since $\mu>4$, it follows that
$\Phi(u)\to-\infty$, as $\|u\|\to\infty$.
(2) Using (H1), we obtain
\[
\Phi(u)\geq \frac{a}{2}\|u\|^2-c_5|u|_p^p-c_6, \quad\forall u\in X,
\]
where $c_5,c_6$ are poisitive constant. Set
\[
\beta_k:=\sup_{v\in Z_k,\, |v\|=1} |v|_p.
\]
Then we obtain
\[
\Phi(u)\geq a\big(\frac{1}{2}-\frac{1}{p}\big)
\big(\frac{c_5}{a}p\beta_k^p\big)^{\frac{2}{2-p}}-c_6
\]
for every $u\in Z_k$ such that
\[
\|u\|=r_k:=\big(\frac{c_5}{a}p\beta_k^p\big)^{\frac{1}{2-p}}.
\]
We know from \cite[Lemma 3.8]{W} that $\beta_k\to0$, as $k\to\infty$.
This implies that $r_k\to\infty$, as $k\to\infty$.
\end{proof}
Now we fix $k$ large enough, and for $m>k+2$, we set
\begin{gather*}
\Phi_m:=\Phi|_{Y_m},\quad
K_m:=\big\{u\in Y_m: \Phi'_m(u)=0\big\},\quad
E_m:=Y_m\backslash K_m, \\
P_m:=\big\{u\in Y_m\,;\,u(x)\geq0\big\},\quad
Z^m_k:=\oplus_{j=k}^m X_j,\quad
N^m_k:=\big\{u\in Z^m_k\,|\,\|u\|=r_k\big\}.
\end{gather*}
We remark that for all $u\in P_m\backslash\big\{0\big\}$ we have
$\int_\Omega ue_1dx>0$, while for all $u\in Z_k$, $\int_\Omega ue_1dx=0$,
where $e_1$ is the principal eigenfunction of the Laplacian.
This implies that $P_m\cap Z_k=\big\{0\big\}$. It then follows, since
$N_k^m$ is compact, that
\begin{equation}\label{deltam}
\delta_m:=\operatorname{dist}\big(N_k^m,-P_m\cup P_m\big)>0.
\end{equation}
For $u\in Y_m$ fixed, we consider the functional
\begin{equation}
I_u(v)=\frac{1}{2}\big(a+b\|u\|^2\big)\int_\Omega|\nabla v|^2dx
-\int_\Omega vf(x,u)dx,\quad v\in Y_m.
\end{equation}
It is not difficult to see that $I_u$ is of class $C^1$, coercive, bounded below,
weakly lower semicontinuous, and strictly convex.
Therefore $I_u$ admits a unique minimizer $v=Au\in Y_m$, which is the unique
solution to the problem
\[
-\big(a+b\|u\|^2\big)\Delta v=f(x,u),\quad v\in Y_m.
\]
Clearly, the set of fixed points of $A$ coincide with $K_m$.
Moreover, the operator $A:Y_m\to Y_m$ has the following important properties.
\begin{lemma}\label{Aop}\quad
\begin{enumerate}
\item[(1)] $A$ is continuous and maps bounded sets to bounded sets.
\item[(2)] For any $u\in Y_m$ we have
\begin{gather}
\langle \Phi_m'(u),u-Au\rangle \geq a\|u-Au\|^2, \label{aa1}\\
\|\Phi_m'(u)\|\leq (a+b)\big(1+\|u\|^2\big)\|u-Au\|.\label{aa2}
\end{gather}
\item[(3)] There exists $\mu_m\in]0,\delta_m[$ such that
$A(\pm D^0_m)\subset \pm D^0_m$, where $\delta_m$ is defined by
\eqref{deltam}.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $(u_n)\subset Y_m$ such that $u_n\to u$. We set $v_n=Au_n$ and $v=Au$.
By the definition of $A$ we have for any $w\in Y_m$,
\begin{gather}
\big(a+b\|u_n\|^2\big)\int_\Omega\nabla v_n\nabla w\,dx
=\int_\Omega wf(x,u_n)dx\label{e1}\\
\big(a+b\|u\|^2\big)\int_\Omega\nabla v\nabla w\,dx
=\int_\Omega wf(x,u)dx.\label{e2}
\end{gather}
Taking $w=v_n-v$ in \eqref{e1} and in \eqref{e2}, and using the H\"{o}lder
inequality and the Sobolev embedding theorem, we obtain
\begin{align*}
&\big(a+b\|u_n\|^2\big)\|v_n-v\|^2\\
&=b\big(\|u_n\|^2-\|u\|^2\big)\int_\Omega\nabla v\nabla(v_n-v)dx
+\int_\Omega(v-v_n)\big(f(u_n)-f(u)\big)dx\\
&\leq c_1\big|\|u_n\|^2-\|u\|^2\big|\|v\|\|v_n-v\|
+c_2\|v_n-v\||f(u_n)-f(u)|_{\frac{p}{p-1}},
\end{align*}
where $c_1,c_2>0$ are constant. By (H1) and \cite[Theorem A.2]{W},
we have $f(u_n)- f(u)\to0$ in $L^{\frac{p}{p-1}}(\Omega)$.
Hence $\|Au_n-Au\|=\|v_n-v\|\to0$, that is, $A$ is continuous.
On the other hand, for any $u\in Y_m$ we have, taking $v=w=Au$ in \eqref{e2}
\[
\big(a+b\|u\|^2\big)\|Au\|^2=\int_\Omega Auf(x,u)dx.
\]
By using (H1), the H\"{o}lder inequality, and the Sobolev embedding theorem,
we obtain
\[
a\|Au\|\leq C\big(1+\|u\|^{p-1}),
\]
where $C>0$ a constant. This shows that $Au$ is bounded whenever $u$ is bounded.
(2) Taking $w=u-Au$ in \eqref{e2}, we obtain
\[
\big(a+b\|u\|^2\big)\int_\Omega\nabla(Au)\nabla(u-Au)dx=\int_\Omega(u-Au)f(x,u)dx,
\]
which implies
\[
\langle \Phi_m'(u),u-Au\rangle =\big(a+b\|u\|^2\big)\|u-Au\|^2\geq a\|u-Au\|^2.
\]
On the other hand, using \eqref{e2}, we obtain
\begin{align*}
\langle \Phi'_m(u),w\rangle
&=\big(a+b\|u\|^2\big)\int_\Omega\nabla u\nabla wdx-\int_\Omega wf(x,u)dx\\
&=\big(a+b\|u\|^2\big)\int_\Omega\nabla (u-Au)\nabla wdx,\quad\forall w\in Y_m.
\end{align*}
This implies
\[
\|\Phi'_m(u)\|\leq \big(a+b\|u\|^2\big)\|u-Au\|.
\]
(3) It follows from (H1) and (H2) that
for each $\varepsilon>0$ there exists $c_\varepsilon>0$ such that
\begin{equation}\label{feps}
|f(x,t)|\leq \varepsilon|t|+c_\varepsilon|t|^{p-1},\quad\forall t\in\mathbb{R}.
\end{equation}
Let $u\in Y_m$ and let $v=Au$. As usual we denote $w^\pm=\max\{0,\pm w\}$,
for any $w\in X$.
Taking $w=v^+$ in \eqref{e2} and using the H\"{o}lder inequality, we obtain
\[
\big(a+b\|u\|^2\big)\|v^+\|^2=\int_\Omega v^+f(x,u)dx
\leq \varepsilon|u^+|_2|v^+|_2+c_\varepsilon|u^+|_p^{p-1}|v^+|_p,
\]
which implies
\begin{equation}\label{v}
\|v^+\|^2\leq \frac{1}{a}\Big(\varepsilon|u^+|_2|v^+|_2
+c_\varepsilon|u^+|_p^{p-1}|v^+|_p\Big).
\end{equation}
On the other hand it is not difficult to see that $|u^+|_q\leq |u-w|_q$
for all $w\in -P_m$ and $1\leq q\leq 2^\star$. Hence there is a constant
$c_1=c_1(q)>0$ such that $|u^+|_q\leq c_1 \operatorname{dist}(u,-P_m)$.
It is obvious that $\operatorname{dist}(v,-P_m)\leq \|v^+\|$.
So we deduce from \eqref{v} and the Sobolev embedding theorem that
\begin{align*}
\operatorname{dist}(v,-P_m)\|v^+\|
&\leq \|v^+\|^2\\
&\leq c_2\Big(\varepsilon \operatorname{dist}(u,-P_m)
+c_\varepsilon \operatorname{dist}(u,-P_m)^{p-1}\Big)\|v^+\|,
\end{align*}
where $c_2>0$ is constant. This implies
\[
\operatorname{dist}(v,-P_m)
\leq c_2\Big(\varepsilon \operatorname{dist}(u,-P_m)
+c_\varepsilon \operatorname{dist}(u,-P_m)^{p-1}\Big).
\]
Similarly one can show that
\[
\operatorname{dist}(v,P_m)\leq c_3\Big(\varepsilon \operatorname{dist}(u,P_m)
+c_\varepsilon \operatorname{dist}(u,P_m)^{p-1}\Big),
\]
for some constant $c_3>0$.
Choosing $\varepsilon$ small enough, we can then find
$\mu_m\in]0,\delta_m[$ such that
\[
\operatorname{dist}(v,\pm P_m)\leq \frac{1}{2} \operatorname{dist}(u,\pm P_m)
\]
whenever $\operatorname{dist}(u,\pm P_m)<\mu_m$.
\end{proof}
Using the $\mu_m$ obtained above, we define
\begin{gather*}
\pm D_m^0:=\big\{u\in Y_m: \operatorname{dist}\big(u,\pm P_m\big)<\mu_m\big\},\\
D_m=D_m^0\cup(-D_m^0), \quad
S_m:=Y_m\backslash D_m.
\end{gather*}
\begin{remark}\label{a2} \rm
Note that $\mu_m<\delta_m$ implies $N_k^m\subset S_m$.
\end{remark}
The vector field $A:Y_m\to Y_m$ does not satisfy the assumption (H7) of
Theorem \ref{scft} as it is not locally Liptschitz continuous.
However, it will be used in the spirit of \cite{BL04} to construct a
vector field which will satisfy the above mentioned condition.
\begin{lemma}\label{Bop}
There exists an odd locally Lipschitz continuous operator $B:E_m\to Y_m$
such that
\begin{itemize}
\item[(1)] $\langle \Phi'(u),u-B(u)\rangle \geq \frac{1}{2}\|u-A(u)\|^2$,
for any $u\in E_m$.
\item[(2)] $\frac{1}{2}\|u-B(u)\|\leq \|u-A(u)\|\leq 2\|u-B(u)\|$,
for any $u\in E_m$.
\item[(3)] $B\big((\pm D^0_m)\cap E_m\big)\subset \pm D^0_m$.
\end{itemize}
\end{lemma}
The proof of this lemma follows the lines of \cite{BL04}.
We provide a sketch of the proof here for completeness.
\begin{proof}
We define $\Delta_1,\Delta_2:E_m\to\mathbb{R}$ as
\begin{equation}\label{deltazero}
\Delta_1(u)=\frac{1}{2}\|u-Au\|\quad \text{and}\quad
\Delta_2(u)=\frac{a}{2(a+b)}(1+\|u\|^2)^{-1}\|u-Au\|.
\end{equation}
For any $u\in E_m$ we choose $\gamma(u)>0$ such that
\begin{equation}\label{deltaun}
\|A(v)-A(w)\|<\min\big\{\Delta_1(v),\Delta_1(w),\Delta_2(v),\Delta_2(w)\big\}
\end{equation}
holds for every $v,w\in N(u):=\big\{z\in Y_m\,;\,\|z-u\|<\gamma(u)\big\}$.
Let $\mathcal{V}$ be a locally finite open refinement of
$\big\{N(u)\,;\,u\in E_m\big\}$ and define
\begin{gather*}
\mathcal{V}^\star:=\big\{V\in\mathcal{V}:
D_m^0\cap V\neq\emptyset,\,-D_m^0\cap V\neq\emptyset,
\,-D_m^0\cap D_m^0\cap V\neq\emptyset \big\},
\\
\mathcal{U}:=\bigcup_{V\in\mathcal{V}\backslash\mathcal{V}^\star}
\big\{V\big\}\cup\bigcup_{V\in\mathcal{V}^\star}
\big\{V\backslash D_m^0,V\backslash(-D_m^0)\big\}.
\end{gather*}
By construction $\mathcal{U}$ is a locally finite open refinement of
$\big\{N(u): u\in E_m\big\}$ and has a property that any
$U\in\mathcal{U}$ is such that
\begin{equation}\label{etoile}
U\cap D_m^0\neq\emptyset\text{ and }U\cap (-D_m^0)
\neq\emptyset\Longrightarrow U\cap D_m^0\cap (-D_m^0)\neq\emptyset.
\end{equation}
Let $\big\{\Pi_U:U\in\mathcal{U}\big\}$ be the partition of unity subordinated
to $\mathcal{U}$ defined by
\[
\Pi_U(u):=\frac{\alpha_U(u)}{\sum_{v\in\mathcal{U}}\alpha_U(v)},
\text{ where }\alpha_U(u)=\operatorname{dist}\big(u,E_m\backslash U\big).
\]
For any $u\in\mathcal{U}$ choose $a_U$ such that if
$U\cap(\pm D_m^0)\neq\emptyset$ then $a_U\in U\cap(\pm D_m^0)$
\big(such an element exists in view of \eqref{etoile}\big).
Define $B:E_m\to Y_m$ by
\[
B(u):=\frac{1}{2}\big(H(u)-H(-u)\big),\quad\text{where }
H(u)=\sum_{U\in\mathcal{U}}\Pi_U(u)A(a_U).
\]
We then conclude as in \cite{BL04} by using Lemma \ref{Aop}-(3),
\eqref{deltazero}, \eqref{deltaun}, and \eqref{aa1}.
\end{proof}
\begin{remark}\label{rema} \rm
Lemmas \ref{Aop} and \ref{Bop} imply that
\begin{gather*}
\langle \Phi_m'(u),u-B(u)\rangle
\geq\frac{1}{8}\|u-B(u)\|^2 \text{ and}\\
\|\Phi_m'(u)\|\leq2(a+b)(1+\|u\|^2)\|u-B(u)\|,
\end{gather*}
for all $u\in E_m$.
\end{remark}
\begin{lemma}\label{tech}
Let $c0$. For all $u\in Y_m$ such that $\Phi_m(u)\in[c,d]$
and $\|\Phi'_m(u)\|\geq\alpha$, there exists $\beta>0$ such that
$\|u-B(u)\|\geq\beta$.
\end{lemma}
\begin{proof}
By the definition of the operator $A$, we have for any $u\in Y_m$,
\[
\big(a+b\|u\|^2\big)\int_\Omega\nabla(Au)\nabla u dx=\int_\Omega uf(x,u)dx.
\]
It follows that
\begin{align*}
&\Phi_m(u)-\frac{1}{\mu}\big(a+b\|u\|^2\big)
\int_\Omega\nabla u\nabla (u-Au) dx\\
&=a\big(\frac{1}{2}-\frac{1}{\mu}\big)\|u\|^2
+b\big(\frac{1}{4}-\frac{1}{\mu}\big)\|u\|^4
+\int_\Omega\big(\frac{1}{\mu}uf(x,u)-F(x,u)\big)dx
\end{align*}
which implies, using (H3) and Lemma \ref{Bop}-(2), that
\begin{equation}
\begin{aligned}
b\big(\frac{1}{4}-\frac{1}{\mu}\big)\|u\|^4
&\leq|\Phi_m(u)|+\frac{1}{\mu}\big(a+b\|u\|^2\big)\|u\|\|u-Au\|\\
&\leq |\Phi_m(u)|+\frac{2}{\mu}\big(a+b\|u\|^2\big)\|u\|\|u-Bu\|.
\end{aligned}\label{deuxetoile}
\end{equation}
Suppose that there exists a sequence $(u_n)\subset Y_m$ such that:
$\Phi_m(u_n)\in[c,d]$, $\|\Phi_m'(u_n)\|\geq\alpha$ and
$\|u_n-Bu_n\|\to0$. By \eqref{deuxetoile} we see that $(\|u_n\|)$ is bounded.
It follows from Remark \ref{rema} above that $\Phi'_m(u_n)\to0$, which is a
contradiction.
\end{proof}
Now we verify the compactness condition for $\Phi$.
\begin{lemma}\label{psnod}
$\Phi$ satisfies the $(PS)^\star_{nod}$ condition, that is:
\begin{itemize}
\item any Palais-Smale sequence of $\Phi_m$ is bounded,
\item any sequence $(u_{m_j})\subset X$ such that:
$m_j\to\infty$, $u_{m_j}\in V_{\frac{\mu_{m_j}}{2}}(S_{m_j})$, \\
$\sup\Phi(u_{m_j})<\infty$, and $\Phi'_{m_j}(u_{m_j})=0$,
has a subsequence converging to a sign-changing critical point of $\Phi$.
\end{itemize}
\end{lemma}
\begin{proof}
For any $u\in Y_m$ we have, in view of (H3),
\begin{equation}
\begin{aligned}
&\Phi_m(u)-\frac{1}{\mu}\langle \Phi_m'(u),u\rangle \\
&=a\big(\frac{1}{2}-\frac{1}{\mu}\big)\|u\|^2
+b\big(\frac{1}{4}-\frac{1}{\mu}\big)\|u\|^4
+\int_\Omega\big(\frac{1}{\mu}uf(x,u)-F(x,u)\big)dx\\
&\geq a\big(\frac{1}{2}-\frac{1}{\mu}\big)\|u\|^2
+b\big(\frac{1}{4}-\frac{1}{\mu}\big)\|u\|^4.
\end{aligned} \label{pss}
\end{equation}
It then follows that any sequence $(u_n)\subset Y_m$ such that
$\sup_n\Phi_m(u_n)<\infty$ and $\Phi_m'(u_n)\to0$ is bounded.
Now let $(u_{m_j})\subset X$ be such that
\[
m_j\to\infty,\quad u_{m_j}\in V_{\frac{\mu_{m_j}}{2}}(S_{m_j}),\quad
\sup\Phi(u_{m_j})<\infty, \quad \Phi'_{m_j}(u_{m_j})=0.
\]
In view of \eqref{pss} the sequence $(u_{m_j})$ is bounded. Hence, up to
a subsequence, $u_{m_j}\rightharpoonup u$ in $X$ and $u_{m_j}\to u$
in $L^p(\Omega)$.
Observe that the condition $\Phi_{m_j}'(u_{m_j})=0$ is weaker than
$\Phi'(u_{m_j})=0$. Therefore, the fact that $(u_{m_j})$ converges strongly,
up to a subsequence, to $u$ in $X$ does not follow from the usual standard argument.
Let us denote by $\Pi_{m_j}:X\to Y_{m_j}$ the orthogonal projection.
Then it is clear that $\Pi_{m_j}u\to u$ in $X$, as $m_j\to\infty$. We have
\begin{equation} \label{eee}
\begin{aligned}
&\langle \Phi'_{m_j}(u_{m_j}),u_{{m_j}}-\Pi_{m_j}u\rangle \\
&=\big(a+b\|u_{m_j}\|^2\big)\langle u_{m_j},u_{m_j}-\Pi_{m_j}u\rangle
-\int_\Omega \big(u_{m_j}-\Pi_{m_j}u\big)f(x,u_{m_j})dx.
\end{aligned}
\end{equation}
Since $(u_{m_j})$ is bounded, we deduce from (H1) that
$\big(|f(x,u_{m_j})|_{p/p-1}\big)$ is bounded. We then obtain by using
the H\"{o}lder inequality
\[
\big|\int_\Omega \big(u_{m_j}-\Pi_{m_j}u\big)f(x,u_{m_j})dx\big|
\leq|u_{m_j}-\Pi_{m_j}u|_p|f(x,u_{m_j})|_{\frac{p}{p-1}}\to0.
\]
Recalling that $\Phi'_{m_j}(u_{m_j})=0$, we deduce from \eqref{eee} that
\[
\langle u_{m_j},u_{m_j}-\Pi_{m_j}u\rangle
=\|u_{m_j}\|^2-\langle u_{m_j},u\rangle
+\langle u_{m_j},u-\Pi_{m_j}u\rangle =\circ(1).
\]
It then follows that $\|u_{m_j}\|\to \|u\|$ which implies, since $X$
is uniformly convex, that $u_{m_j}\to u$ in $X$. It is readily seen that
$u$ is a critical point of $\Phi$.
To show that the limit $u$ is sign-changing, we first observe that
\begin{align*}
\langle \Phi'_{m_j}(u_{m_j}),u_{m_j}^\pm\rangle =0\;
&\Leftrightarrow\; \big(a+b\|u_{m_j}\|^2\big)\|u_{m_j}^\pm\|^2
=\int_\Omega u_{m_j}^\pm f(x,u_{m_j})dx\\
&\Rightarrow\; a\|u_{m_j}^\pm\|^2\leq \int_\Omega u_{m_j}^\pm f(x,u_{m_j}^\pm)dx.
\end{align*}
By using \eqref{feps} and the Sobolev embedding theorem, we obtain
\[
a\|u_{m_j}^\pm\|^2\leq \int_\Omega u_{m_j}^\pm f(x,u_{m_j}^\pm)dx
\leq c\big(\varepsilon\|u_{m_j}^\pm\|^2+c_\varepsilon\|u_{m_j}^\pm\|^p\big),
\]
where $c>0$ is a constant. Since $u_{m_j}$ is sign-changing,
$u_{m_j}^\pm$ are not equal to $0$. Choosing $\varepsilon$
small enough \big(for instance $\varepsilon<\frac{a}{2c}$\big),
we see that $(\|u_{m_j}^\pm\|)$ are bounded below by strictly positive
constants which do not depend on $m_j$. This implies that the limit $u$
of the sequence $(u_{m_j})$ is also sign-changing.
\end{proof}
We are now in a position for proving our main result.
\begin{proof}[Proof Theorem \ref{mainresult}]
By Lemmas \ref{akbk}, \ref{Bop}, \ref{tech}, and \ref{psnod}, and
Remarks \ref{a2} and \ref{rema}, conditions (H5), (H6), (H7) and (H8)
of Theorem \ref{scft} are satisfied.
It then suffices to apply Theorem \ref{scft} to conclude.
\end{proof}
\subsection*{Acknowledgements}
We express our warm gratitude to the anonymous referee for comments which
contributed to a significant improvement of the paper.
This research was supported by The Fields Institute for Research in
Mathematical Sciences and The Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the
Ministry of Research and Innovation.
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