Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 56, pp. 1-16.

Title: Existence of weak solutions for quasilinear elliptic equations 
       involving the p-Laplacian
 
Author: Uberlandio Severo (Univ. Federal da Paraiba, Brazil)

Abstract: 
 This paper shows the existence of nontrivial weak
 solutions for the quasilinear elliptic equation
 $$
 -\big(\Delta_p u +\Delta_p (u^2)\big) +V(x)|u|^{p-2}u= h(u)
 $$
 in $\mathbb{R}^N$. Here $V$ is a positive continuous
 potential bounded away from zero and $h(u)$ is a nonlinear term
 of subcritical type.
 Using minimax methods, we show the existence of a nontrivial solution
 in $C^{1,\alpha}_{\rm loc}(\mathbb{R}^N)$ and then show that it
 decays to zero at infinity when $1<p<N$.
 
Submitted October 27, 2007. Published April 17, 2008.
Math Subject Classifications: 35J20, 35J60, 35Q55.
Key Words: Quasilinear Schrodinger equation; solitary waves;
           p-Laplacian; variational method; mountain-pass theorem.