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Existence and concentration phenomena for a class of indefinite variational problems with critical growth. (arXiv:1801.08138v1 [math.AP])
来源于:arXiv
In this paper we are interested to prove the existence and concentration of
ground state solution for the following class of problems $$ -\Delta
u+V(x)u=A(\epsilon x)f(u), \quad x \in \R^{N}, \eqno{(P)_{\epsilon}} $$ where
$N \geq 2$, $\epsilon>0$, $A:\R^{N}\rightarrow\R$ is a continuous function that
satisfies $$
0<\inf_{x\in\R^{N}}A(x)\leq\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x\in\R^{N}}A(x)=A(0),\eqno{(A)}
$$ $f:\R\rightarrow\R$ is a continuous function having critical growth,
$V:\R^{N}\rightarrow\R$ is a continuous and $\Z^{N}$--periodic function with
$0\notin\sigma(\Delta+V)$. By using variational methods, we prove the existence
of solution for $\epsilon$ small enough. After that, we show that the maximum
points of the solutions concentrate around of a maximum point of $A$. 查看全文>>