Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions
Francisco Odair de Paiva Humberto Ramos Quoirin
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1219-1227 doi: 10.3934/dcds.2009.25.1219
We study multiplicity of solutions for a quasilinear elliptic problem related to the $p$-Laplacian operator. Our assumptions rely on the first eigenvalue depending on a weight function. We treat both resonant and non-resonant cases.
keywords: p-Laplacian multiplicity of solution. indefinite weight
Multiple solutions for a class of quasilinear problems
Francisco Odair de Paiva
Discrete & Continuous Dynamical Systems - A 2006, 15(2): 669-680 doi: 10.3934/dcds.2006.15.669
In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem

$-\Delta_p u = g(x,u)$  in  $\Omega$
$u = 0 $  on  $\partial \Omega$,

where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function such that $g(x,0)=0$ and which is asymptotically linear. We suppose that $g(x,t)/t$ tends to an $L^r$-function, $r>N/p$ if 1 < p ≤ N and $r=1$ if $p>N$, which can change sign. We consider both the resonant and the nonresonant cases.

keywords: indefinite weight Quasilinear problems multiplicity of solutions.
A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems
Lynnyngs Kelly Arruda Francisco Odair de Paiva Ilma Marques
Conference Publications 2011, 2011(Special): 112-116 doi: 10.3934/proc.2011.2011.112
Using variational methods, we prove a nonexistence and multiplicity result of positive solutions for a class of elliptic systems involving a parameter.
keywords: (p nonexistence multiplicity positive solutions q)-Laplacian problems variational methods

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