Existence and asymptotic behavior of solutions for quasilinear Schrödinger equations involving p-Laplacian

Abstract

In this paper, we investigate the existence and asymptotic behavior of positive solutions for quasilinear Schrödinger equations involving p-Laplacian

$$\begin{aligned} -\Delta _{p}u + \kappa \Delta _{p}(u^2)u + (\lambda A( x) + 1)|u|^{p-2}u = h(u), \quad u\in W^{1,p}(\mathbb {R}^N), \end{aligned}$$

where \(2<p<N\), \(\kappa ,\) \(\lambda \) are parameters and A(x) is a potential. The problem is quite sensitive to the sign of \(\kappa \) and there have been many results for \(\kappa \le 0.\) By means of minimization on the Nehari manifold together with perturbation type techniques, we establish the existence of positive solutions for small \(\kappa >0\) and large \(\lambda \). Moreover, we show that the solutions \(u_{\kappa ,\lambda }\) converge in \(W^{1,p}\) to a positive solution of p-Laplacian in a bounded domain as \((\kappa ,\lambda )\rightarrow (0^+,+\infty )\). Our results extend some known results of \(\kappa \le 0\).