Elliptic problem driven by different types of nonlinearities

TL;DR

This paper proves the existence and multiplicity of solutions for a fractional elliptic problem involving nonlinearities with different growth types, including exponential critical growth, using variational methods.

Contribution

It establishes new existence and multiplicity results for a fractional elliptic problem with combined nonlinearities, including logarithmic convolution and exponential growth.

Findings

Existence of a nontrivial solution at mountain pass level.

Existence of a ground state solution under exponential critical growth.

Solutions are obtained using variational methods and critical point theory.

Abstract

In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R}, \end{split} \end{align*} where $\mu>0$, $(*)$ is the convolution operation between two functions, $0<\gamma<1$, $f$ is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity $f$ is of exponential critical growth.