Lack of coercivity for N-Laplace equation with critical exponential nonlinearities in a bounded domain

Abstract

In this article, we study the existence and multiplicity of non-negative solutions of the $N$-Laplacian equation $$\displaylines{ -\Delta_N u+V(x)|u|^{N-2}u = \lambda h(x)|u|^{q-1}u+ u|u|^{p} e^{|u|^{\beta}} \quad \text{in }\Omega \cr u \geq 0 \quad \text{in } \Omega,\quad u\in W^{1,N}_0(\Omega),\cr u =0 \quad \text{on } \partial \Omega }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq 2$, $00$. By minimization on a suitable subset of the Nehari manifold, and using fiber maps, we find conditions on $V$, $h$ for the existence and multiplicity of solutions, when $V$ and $h$ are sign changing and unbounded functions.

Keywords

• Quasilinear problem
• critical exponent
• Trudinger-Moser embedding
• sign-changing weight function