On the existence and multiplicity of solutions for the $N$-Choquard logarithmic equation with exponential critical growth

Abstract

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution, for the critical case, and the existence of infinitely many solutions, for the subcritical case, to the Choquard Logarithmic equation, $-{\Delta }_{N}u+a(x)|u{|}^{N-2}u+\lambda (\ln |\cdot |\ast |u{|}^N)|u{|}^{N-2}u=f(u)\text{ in }{\mathbb{R}}^N$, where $a:{\mathbb{R}}^N\to \mathbb{R}$, $\lambda >0$, $N\ge 3$ and $f:\mathbb{R}\to [0,\infty )$ is continuous function that behaves like $\exp (\alpha |u{|}^{\frac{N}{N-1}})$ at infinity, for $\alpha >0$.

12 pages

References in corpus (2)

• Bound states of two-dimensional Schrödinger-Newton equations
• Existence and multiplicity of solutions for the fractional $p$-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth