Electron. J. Diff. Equ., Vol. 2014 (2014), No. 15, pp. 1-22.

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

Sarika Goyal, Konijeti Sreenadh

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 $\mathbb{R}^N$, $N\geq 2$, $0< q<N-1 <p+1$, $\beta\in (1,\frac{N}{N-1}]$ and $\lambda>0$. 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 8, 2014.
Math Subject Classifications: 35J35, 35J60, 35J92.
Key Words: Quasilinear problem; critical exponent; Trudinger-Moser embedding; sign-changing weight function.

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Sarika Goyal
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: sarika1.iitd@gmail.com}
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: sreenadh@maths.iitd.ac.in

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