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

TL;DR

This paper proves the existence of nonnegative ground state solutions for a class of quasilinear Schrödinger equations involving anisotropic p-Laplacian operators with Minkowski norms, under specific conditions on the potential function.

Contribution

It establishes the existence of ground state solutions for a complex anisotropic quasilinear Schrödinger equation involving the Finsler p-Laplacian, extending previous results to more general operators.

Findings

Existence of non-trivial non-negative bounded ground state solutions.

Conditions on the potential function V ensure solution existence.

Application of variational methods to anisotropic p-Laplacian equations.

Abstract

This paper is concerned with the existence of a nonnegative ground state solution of the following quasilinear Schr\"{o}dinger equation \begin{equation*} \begin{split} -\Delta_{H,p}u+V(x)|u|^{p-2}u-\Delta_{H,p}(|u|^{2\alpha}) |u|^{2\alpha-2}u=\lambda |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 $N\geq2$; $(\alpha,p)\in D_N=\{(x,y)\in \;R^2 : 2xy\geq y+1,\; y\geq2x,\; y<N\}$ 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.