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

In this paper, we investigate the existence and asymptotic behavior of positive solutions for quasilinear Schrödinger equations involving p-Laplacian \(\begin{aligned} -\Delta _{p}u + \kappa \Delta _{p}(u^2)u + (\lambda A( x) + 1)|u|^{p-2}u = h(u), \quad u\in W^{1,p}(\mathbb {R}^N), \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u+\kappa {\mathrm{\Delta }}_p(u^2)u+(\lambda A(x)+1){|u|}^{p-2}u=h(u),u\in W^{1,p}({\mathbb{R}}^N),\end{array}$$ where \(2<p<N\) $2<p<N$ , \(\kappa ,\) $\kappa ,$ \(\lambda \) $\lambda$ are parameters and A(x) is a potential. The problem is quite sensitive to the sign of \(\kappa \) $\kappa$ and there have been many results for \(\kappa \le 0.\) $\kappa \le 0.$ By means of minimization on the Nehari manifold together with perturbation type techniques, we establish the existence of positive solutions for small \(\kappa >0\) $\kappa >0$ and large \(\lambda \) $\lambda$ . Moreover, we show that the solutions \(u_{\kappa ,\lambda }\) $u_{\kappa ,\lambda }$ converge in \(W^{1,p}\) $W^{1,p}$ to a positive solution of p-Laplacian in a bounded domain as \((\kappa ,\lambda )\rightarrow (0^+,+\infty )\) $(\kappa ,\lambda )\to (0^{+},+\infty )$ . Our results extend some known results of \(\kappa \le 0\) $\kappa \le 0$ .