Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent

TL;DR

This paper establishes the existence of ground state solutions for a class of nonlinear Choquard equations with critical exponent perturbations using variational methods, addressing challenges posed by the lower critical exponent.

Contribution

It introduces new variational techniques to prove ground state solutions for Choquard equations with critical nonlinear perturbations.

Findings

Existence of ground state solutions proven

Handling of critical exponent in nonlinear Choquard equations

Extension of variational methods to nonlocal equations with perturbations

Abstract

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$, the exponent $\frac{\alpha}{N}+1$ is critical with respect to the Hardy--Littlewood--Sobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions.