Abstract:

In this paper, we mainly study the following nonlinear Schrödinger equation with variable exponents

$\begin{equation} \left\{ \begin{aligned} &-\varepsilon^2\Delta u+V(y)u=|u|^{p(y)-1}u,\ \ \ u\in\mathbb{R}^N,\\ &u(y)\rightarrow 0,\ \ \ |y|\rightarrow +\infty, \end{aligned} \right. \nonumber \end{equation}$

where $\varepsilon>0$ is a sufficiently small parameter, the spatial dimension $N\geq3$, the potential function $V(y)$ satisfies $0, and the variable exponent function $p(y)$ satisfies $1($2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent). By employing the Lyapunov-Schmidt reduction method, we prove that for any positive integer $k$, when $\varepsilon>0$ is sufficiently small, there exists a sharp peak solution to the equation with $k$ peaks, and these $k$ peaks are concentrated at the $k$ critical points of the potential function $V(y)$ as $\varepsilon \rightarrow 0$, respectively.

Key words: Lyapunov-Schmidt reduction method, Schrödinger equation, variable exponent, spiked solutions, multiplicity