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

Abstract

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -Δu+ u=\big(I_α*|u|^{\fracα{N}+1}\big)|u|^{\fracα{N}-1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\ge 1$, $I_{\alpha }$ 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.

arXiv comment18 pages

References in corpus (9)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• A guide to the Choquard equation
• Existence of groundstates for a class of nonlinear Choquard equations
• Gravitation and quantummechanical localization of macroobjects
• Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent
• Newtonian Quantum Gravity
• Choquard equations under confining external potentials
• Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent
• Interpolation inequalities between Sobolev and Morrey-Campanato spaces: A common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities