Elliptic problem driven by different types of nonlinearities

Abstract

In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-Δ)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+μ|u|^{-γ-1}u,~\text{in}~\mathbb{R}, \end{split} \end{align*} where $μ>0$, $(*)$ is the convolution operation between two functions, $0<γ<1$, $f$ 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.