Electron. J. Diff. Eqns., Vol. 2008(2008), No. 56, pp. 1-16.

Existence of weak solutions for quasilinear elliptic equations involving the p-Laplacian

Uberlandio Severo

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 $C^{1,\alpha}_{\hbox{loc}}(\mathbb{R}^N)$ 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.

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Uberlandio Severo
Departamento de Matemática, Universidade Federal da Paraí ba
58051-900, João Pessoa, PB , Brazil
email: uberlandio@mat.ufpb.br  Tel. 55 83 3216 7434   Fax 55 83 3216 7277

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