Semi-classical states for a class of quasilinear Schrödinger equations with a parameter

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Abstract

In this paper, we study the following quasilinear Schrödinger equation with a parameter: $-\varepsilon^{2} \Delta u+V(x) u-\varepsilon^{2} \kappa \beta \Delta\left(|u|^{2 \beta}\right)|u|^{2 \beta-2} u=|u|^{q-2} u+|u|^{(2 \beta) 2^{*}-2} u \text { in } \mathbb{R}^{N}$, where $N \geqslant 3, \beta>\frac{1}{2}, 4 \beta<q<(2 \beta) 2^{*}$, κ is a constant, and V : $\mathbb{R}^{N} \rightarrow \mathbb{R}$ satisfies the classical global assumption. By using a change of variable, we obtain the existence, multiplicity, and concentration behavior. We will see that the existence of solutions depends heavily on the parameter β.

Keywords

semi-classical states / quasilinear Schrödinger equations / a parameter / critical growth

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