Positive solutions for a class of generalized quasilinear Schrödinger equation involving concave and convex nonlinearities in Orilicz space

Abstract

In this paper, we study the following generalized quasilinear Schrödinger equation \begin{equation*} -\text{div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2+V(x)u=\lambda f(x,u)+h(x,u),\qquad x\in\mathbb{R}^N, \end{equation*} where $\lambda>0$, $N\geq3$, $g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^{+})$. By using a change of variable, we obtain the existence of positive solutions for this problem with concave and convex nonlinearities via the Mountain Pass Theorem. Our results generalize some existing results.

Keywords

• generalized quasilinear schrödinger equation
• positive solutions
• concave and convex nonlinearities