$n$Kirchhoff type equations with exponential nonlinearities

TL;DR

This paper investigates the existence and multiplicity of non-negative solutions for a class of non-local $n$-Kirchhoff equations with exponential nonlinearities, using variational methods on the Nehari manifold.

Contribution

It introduces a novel approach to analyze $n$-Kirchhoff problems with exponential growth nonlinearities, establishing existence and multiplicity results.

Findings

Existence of solutions under certain conditions.

Multiple solutions when $f$ is concave near zero and convex at infinity.

Application of minimization on the Nehari manifold.

Abstract

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_{\Omega}|\nabla u|^n)\Delta_n u = f(x,u) \; \text{in}\; \Omega,\quad u =0\quad\text{on} \quad \partial \Omega, \end{array} \right.$$ where $\Omega\subset \mathbf{R}^n$ is a bounded domain with smooth boundary, $n\geq 2$ and $f$ behaves like $e^{|u|^{\frac{n}{n-1}}}$ as $|u|\to\infty$. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when $f(x,t)$ is concave near $t=0$ and convex as $t\rightarrow \infty.$