$n$Kirchhoff type equations with exponential nonlinearities

Abstract

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$\{\begin{array}{l}-m({\int }_{\Omega }|\nabla u{|}^n){\Delta }_{n}u=f(x,u)\text{in}\Omega ,u=0\text{on}\partial \Omega ,\end{array}$$ where $\Omega \subset {\mathbf{R}}^n$ is a bounded domain with smooth boundary, $n\ge 2$ 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 $t\to \infty .$

Results from earlier version are improved. RACSAM - Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2015

References in corpus (1)

• Ground state solution for a Kirchhoff problem with exponential critical growth