On the logarithmic Schrodinger equation

TL;DR

This paper develops a variational method to prove the existence of infinitely many solutions and a unique positive radially symmetric solution for a class of logarithmic Schrödinger equations, relevant in physics.

Contribution

It introduces a direct variational approach using nonsmooth critical point theory for semi-linear elliptic equations with logarithmic nonlinearities, establishing new existence and uniqueness results.

Findings

Existence of infinitely many weak solutions.

Existence of a unique positive radially symmetric solution.

Solution nondegeneracy and symmetry properties.

Abstract

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.