December  2002, 1(4): 531-545. doi: 10.3934/cpaa.2002.1.531

Signed solution for a class of quasilinear elliptic problem with critical growth

1. 

Departamento de Matematica, Universidade Federal da Paraiba, 58100-907 Campina Grande-PB, Brazil

2. 

Departamento de Matematica, Universidade Federal de Minas Gerais, 31270-010 Belo Horizonte-MG, Brazil

3. 

Departamento de Matematica, Universidade Federal de Vicosa, 36371-000 Vicosa-MG, Brazil

Received  October 2001 Revised  August 2002 Published  September 2002

In this paper we will study the existence of signed solutions for problems of the type

$-L u=\lambda h(x)|x|^{\delta}(u_{+})^q-|x|^{\gamma}(u_{-})^p, \quad $ in $\Omega$,

$u_{\pm}$ ≠0, $\quad u\in E,$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (P)

where $\Omega$ is either a whole space $\mathbb R^N$ or a bounded smooth domain, $Lu =:$ div$(|x|^{\alpha}|\nabla u|^{m-2}\nabla u), $ $\lambda >0, \quad0 < q < m-1 < p \leq m$*$-1,$ $\alpha, $ $\delta $ and $\gamma $ are real numbers, $ N> m-\alpha, $ $m$*$=\frac{(\gamma+N)m}{(\alpha+N-m)}$, $h:\Omega \rightarrow \mathbb R$ is a positive continuous function, $u_{\pm}=\max \{\pm u,0\}$ and $E$ is a Banach space that will be defined later on. We will show that (P) has a solution that changes sign in several situations. The proof of the main results are done by using variational methods applied to the energy functional associated to $(P)$.

Citation: Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531
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