Existence of weak solutions for quasilinear Schrödinger equations with a parameter

Abstract

In this paper, we study the following quasilinear Schrödinger equation of the form \begin{equation*} -\Delta_{p}u+V(x)|u|^{p-2}u-\left[\Delta_{p}(1+u^{2})^{\alpha/2}\right]\frac{\alpha u}{2(1+u^{2})^{(2-\alpha)/2}}=k(u),\qquad x\in \mathbb{R}^{N}, \end{equation*} where $p$-Laplace operator $\Delta_{p}u={\rm div}(|\nabla u|^{p-2}\nabla u)\ (1<p\leq N)$ and $\alpha\geq1$ is a parameter. Under some appropriate assumptions on the potential $V$ and the nonlinear term $k$, using some special techniques, we establish the existence of a nontrivial solution in $C_{\rm loc}^{1,\beta}(\mathbb{R}^{N})\ (0<\beta<1),$ we also show that the solution is in $L^{\infty}(\mathbb{R}^{N})$ and decays to zero at infinity when $1<p<N$.

Keywords

• quasilinear schrödinger equation
• variational method
• mountain-pass theorem
• $p$-laplace operator