Multiple Solitary Wave Solutions for Nonhomogeneous Quasilinear Schrödinger Equations

Abstract

AbstractWe are concerned with the following generalized quasilinear Schrödinger equations $$\begin{align*}&-\text{div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2+V(x)u=h(u)+\mu k(x), \\&\qquad\quad x\in \mathbb{R}^N,\end{align*}$$where $N\ge 3, \ g:\mathbb{R}\rightarrow\mathbb{R}^+$ is an even differentiable function satisfying $\displaystyle \lim_{t \rightarrow +\infty} \frac {g(t)}{t^{\alpha -1}} = \beta /gt0$ for some $\alpha \ge 1$, h is a nonlinear function covering the case $h(t)=|t|^{p-2}t\ (2\ltp\lt\alpha2^*)$, the potential $V:\mathbb{R}^N\rightarrow\mathbb{R}$ is positive and µk(x) is a perturbation term with µ > 0. Combining the change of variables and variational arguments, we show that the given problem has at least two positive solutions for some $\mu_0\gt0$ and $\mu\in(0,\mu_0)$.