Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 15, pp. 1-22.
Title: Lack of coercivity for N-Laplace equation with critical
exponential nonlinearities in a bounded domain
Authors: Sarika Goyal (Indian Institute of Technology Delhi, New Delhi, India)
Konijeti Sreenadh (Indian Institute of Technology Delhi, New Delhi, India)
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 $\mathbb{R}^N$, $N\geq 2$,
$0< q0$. 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.
Submitted May 23, 2013. Published January 08, 2014.
Math Subject Classifications: 35J35, 35J60, 35J92.
Key Words: Quasilinear problem; critical exponent;
Trudinger-Moser embedding; sign-changing weight function.