Existence and multiplicity results for a class of Kirchhoff-Choquard equations with a generalized sign-changing potential

TL;DR

The paper studies a planar Kirchhoff-Choquard equation in $\mathbb{R}^{2}$ with a sign-changing, potentially unbounded kernel and a nonlinearity of exponential critical growth, posed with $M(t)=a+bt$. A variational framework is developed on the weighted space $X$, leveraging the Moser–Trudinger inequality to handle the exponential growth and nonlocal convolution terms. Under nondegeneracy ($a>0$), the authors establish mountain-pass geometry and prove the existence of a nontrivial solution and a ground-state, with an additional symmetry assumption $f$ yielding infinitely many solutions; in the degenerate case ($a=0$) they obtain existence results for small $\mu$ via concentration-compactness and a mountain-pass construction. The work extends Kirchhoff-Choquard analysis to sign-changing and potentially unbounded potentials in the planar setting, providing both existence and multiplicity results through variational methods and symmetry arguments with careful control of nonlocal terms and exponential nonlinearity.

Abstract

In the present work we are concerned with the following Kirchhoff-Choquard-type equation $$-M(||\nabla u||_{2}^{2})Δu +Q(x)u + μ(V(|\cdot|)\ast u^2)u = f(u) \mbox{ in } \mathbb{R}^2 , $$ for $M: \mathbb{R} \rightarrow \mathbb{R}$ given by $M(t)=a+bt$, $ μ>0 $, $ V $ a sign-changing and possible unbounded potential, $ Q $ a continuous external potential and a nonlinearity $f$ with exponential critical growth. We prove existence and multiplicity of solutions in the nondegenerate case and guarantee the existence of solutions in the degenerate case.

Theorems & Definitions (52)

• Example 1.1
• Example 1.2
• Example 1.3
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Example 2.1
• Lemma 2.3