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\begin{document}

\title{Existence, multiplicity and concentration of positive solutions for a
modified Schr\textbackslash{}''\{o\}dinger equation with critical
exponent}



\author[1]{Xian Wu}%
\author[2]{Xingwei Zhou}%
\affil[1]{Yunnan Normal University}%
\affil[2]{Department of Mathematics and Statistics, Yunnan University}%


\vspace{-1em}



  
  \date{May 28, 2024}


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\begin{abstract}
In this paper, we concern the modified Schr\textbackslash{}``\{o\}dinger
equations \$\$\textbackslash{}
-\{\textbackslash{}varepsilon\}\^{}\{2\}\textbackslash{}Delta
u+V(x)u-\textbackslash{}varepsilon\^{}\{2\}u\textbackslash{}Delta
u\^{}2=\textbar{}u\textbar{}\^{}\{22\^{}*-2\}u+g(u), \textbackslash{} x
\textbackslash{} \textbackslash{}in
\textbackslash{}mathbb\{R\}\^{}N.\$\$ First, a existence result of
ground state positive solutions is given. Next, we research multiplicity
and concentration of positive solutions. Where \$N\textbackslash{}geq
2\$, \$\textbackslash{}varepsilon\$ is positive parameters and
\$2\^{}*=\textbackslash{}frac\{2N\}\{N-2\}\$ is the critical exponent,
\$V \textbackslash{}in C(\textbackslash{}mathbb\{R\}\^{}N,
\textbackslash{}mathbb\{R\^{}\{+\}\})\$, \$g \textbackslash{}in
C(\textbackslash{}mathbb\{R\}, \textbackslash{}mathbb\{R\})\$. Our
results improve corresponding results in (missing citation) (X. He, A.
Qian, W. Zou, Existence and concentration of positive solutions for
quasilinear Schr\textbackslash{}''\{o\}dinger equations with critical
growth, Nonlinearity, 26(2013), 3137-3168).%
\end{abstract}%



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\section*{References}\sloppy
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