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