Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space

TL;DR

This paper proves the existence of positive solutions for a class of quasi-linear elliptic equations with exponential growth in the entire Euclidean space, using variational methods and critical inequalities.

Contribution

It establishes the existence of one or two positive solutions for these equations under certain conditions, combining advanced inequalities and variational principles.

Findings

Existence of a nontrivial positive weak solution.

Existence of two distinct positive weak solutions for a perturbed equation.

Application of Trudinger-Moser inequality and variational methods.

Abstract

In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger-Moser type inequality. Under some assumptions on the potential and the nonlinearity, it is proved that there is a nontrivial positive weak solution to this equation. Also it is shown that there are two distinct positive weak solutions to a perturbation of the equation. The method of proving these results is combining Trudinger-Moser type inequality, Mountain-pass theorem and Ekeland's variational principle.