Ground state solutions for quasilinear Schrodinger type equation involving anisotropic p-laplacian

Abstract

This paper is concerned with the existence of a nonnegative ground state solution of the following quasilinear Schrödinger equation \begin{equation*} \begin{split} -Δ_{H,p}u+V(x)|u|^{p-2}u-Δ_{H,p}(|u|^{2α}) |u|^{2α-2}u=λ|u|^{q-1}u \text{ in }\;R^n;\; u\in W^{1,p}(\;R^n)\cap L^\infty(\;R^N) \end{split} \end{equation*} where $N\ge 2$; $(\alpha ,p)\in D_N=\{(x,y)\in R^2:2xy\ge y+1,y\ge 2x,y<N\}$ and $\lambda >0$ is a parameter. The operator ${\Delta }_{H,p}$ is the reversible Finsler p-Laplacian operator with the function $H$ being the Minkowski norm on $R^N$. Under certain conditions on $V$, we establish the existence of a non-trivial non-negative bounded ground state solution of the above equation.

32 pages