$(p,N)-$Choquard logarithmic equation involving a nonlinearity with exponential critical growth: existence and multiplicity

Abstract

The present work is concerned with the following version of Choquard Logarithmic equations $-{\Delta }_{p}u-{\Delta }_{N}u+a|u{|}^{p-2}u+b|u{|}^{N-2}u+\lambda (\ln |\cdot |\ast G(u))g(u)=f(u)\text{ in }{\mathbb{R}}^N$ , where $a,b,\lambda >0$, $\max \{\frac{N}{2},2\}<p<N$, $f,g:\mathbb{R}\to \mathbb{R}$ are continuous functions that behave like $\exp (\alpha |u{|}^{\frac{N}{N-1}})$ at infinity, for $\alpha >0$, and that has polynomial growth, respectively, and $G(s)=\underset{0}{\overset{s}{\int }}g(\tau )d\tau$. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, using a version of the Symmetric Mountain-Pass Theorem, we get infinitely many solutions.

21 pages