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\begin{document}

\title{MULTIPLICITY OF SOLUTIONS TO CLASS OF NONLOCAL ELLIPTIC PROBLEMS WITH
CRITICAL EXPONENTS}



\author[1]{Elard Hurtado}%
\author[2]{OLIMPIO MIYAGAKI}%
\author[3]{Rodrigo Rodrigues}%
\affil[1]{Universidade de Brasilia}%
\affil[2]{Universidade Federal de São Carlos Centro de Ciências Exatas e de Tecnologia}%
\affil[3]{Universidade Federal de Sao Carlos}%


\vspace{-1em}



  
  \date{May 04, 2024}


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\selectlanguage{english}
\begin{abstract}
In this paper, we establish existence of infinitely many weak solutions
for a class of quasilinear stationary Kirchhoff type equations, which
involves a general variable exponent elliptic operator with critical
growth. Precisely, we study the following nonlocal problem
\textbackslash{}begin\{equation*\} \textbackslash{}begin\{cases\}
-\textbackslash{}displaystyle\{M\}(\textbackslash{}mathscr\{A\}(u))\textbackslash{}operatorname\{div\}\textbackslash{}Bigl(a(\textbar{}\textbackslash{}nabla
u\textbar{}\^{}\{p(x)\})\textbar{}\textbackslash{}nabla
u\textbar{}\^{}\{p(x)-2\}\textbackslash{}nabla u\textbackslash{}Bigl) =
\textbackslash{}lambda f(x,u)+ \textbar{}u\textbar{}\^{}\{s(x)-2\}u
\textbackslash{}text\{ in \}\textbackslash{}Omega,
\textbackslash{}\textbackslash{} u = 0 \textbackslash{}text\{ on \}
\textbackslash{}partial \textbackslash{}Omega,
\textbackslash{}end\{cases\} \textbackslash{}end\{equation*\} where
\$\textbackslash{}Omega\$ is a bounded smooth domain of
\$\textbackslash{}mathbb\{R\}\^{}N,\$ with homogeneous Dirichlet
boundary conditions on \$\textbackslash{}partial
\textbackslash{}Omega,\$ the nonlinearity
\$f:\textbackslash{}overline\{\textbackslash{}Omega\}\textbackslash{}times
\textbackslash{}mathbb\{R\}\textbackslash{}to
\textbackslash{}mathbb\{R\}\$ is a continuous function,
\$a:\textbackslash{}mathbb\{R\}\^{}\{+\}\textbackslash{}to\textbackslash{}mathbb\{R\}\^{}\{+\}\$
is a function of the class \$C\^{}\{1\},\$
\$M:\textbackslash{}mathbb\{R\}\^{}\{+\}\_\{0\}\textbackslash{}to\textbackslash{}mathbb\{R\}\^{}\{+\}\$
is a continuous function, whose properties will be introduced later,
\$\textbackslash{}lambda\$ is a positive parameter and
\$p,s\textbackslash{}in
C(\textbackslash{}overline\{\textbackslash{}Omega\})\$. We assume that
\$\textbackslash{}mathscr\{C\}=\textbackslash{}\{x\textbackslash{}in
\textbackslash{}Omega:
s(x)=\textbackslash{}gamma\^{}\{*\}(x)\textbackslash{}\}\textbackslash{}neq
\textbackslash{}emptyset,\$ where
\$\textbackslash{}gamma\^{}\{*\}(x)=N\textbackslash{}gamma(x)/(N-\textbackslash{}gamma(x))\$
is the critical Sobolev exponent. We will prove that the problem has
infinitely many solutions and also we obtain the asymptotic behavior of
the solution as \$\textbackslash{}lambda\textbackslash{}to 0\^{}\{+\}\$.
Furthermore, we emphasize that a difference with previous researches is
that the conditions on \$a(\textbackslash{}cdot)\$ are general overall
enough to incorporate some interesting differential operators. Our work
covers a feature of the Kirchhoff's problems, that is, the fact that the
Kirchhoff's function \$M\$ in zero is different from zero, it also
covers a wide class of nonlocal problems for \$p(x)\textgreater{}1,\$
for all \$x\textbackslash{}in
\textbackslash{}overline\{\textbackslash{}Omega\}.\$ The main tool to
find critical points of the Euler Lagrange functional associated with
this problem is through a suitable truncation argument,
concentration-compactness principle for variable exponent found in
(missing citation), and the genus theory introduced by Krasnoselskii.%
\end{abstract}%



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\textbf{Hosted file}

\verb`paper July.tex`  available at \url{https://authorea.com/users/428391/articles/532248-multiplicity-of-solutions-to-class-of-nonlocal-elliptic-problems-with-critical-exponents}

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\section*{References}\sloppy
\end{document}

