Existence of positive solutions for a class of $p$-Laplacian type generalized quasilinear Schrödinger equations with critical growth and potential vanishing at infinity

Abstract

In this paper, we study the existence of positive solutions for the following generalized quasilinear Schrödinger equation \begin{equation*} -\operatorname{div}(g^p(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g'(u)|\nabla u|^p+V(x)|u|^{p-2}u =K(x)f(u)+Q(x)g(u)|G(u)|^{p^*-2}G(u),\qquad x\in\mathbb R^N, \end{equation*} where $N\geq 3$, $1<p\leq N$, $p^*=\frac{Np}{N-p}$, $g\in\mathcal{C}^1(\mathbb R,\mathbb R^{+})$, $V(x)$ and $K(x)$ are positive continuous functions and $G(u)=\int_0^ug(t)dt$. By using a change of variable, we obtain the existence of positive solutions for this problem by using the Mountain Pass Theorem. Our results generalize some existing results.

Keywords

• generalized quasilinear schrödinger equation
• positive solutions
• critical growth
• $p$-laplacian