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On some asymmetric elliptic problems with indefinite weights. (Sur certains problèmes elliptiques asymétriques avec poids indéfinis.) (French. English summary) Zbl 0973.35147
Summary: We prove the existence of a first nontrivial eigenvalue for the problem \[ -\Delta_p u= \lambda[m(x)(u^+)^{p- 1}- n(x)(u^-)^{p- 1}]\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega. \] As applications we study the solvability of \[ -\Delta_p u= f(x, u)\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega \] as well as the Fučik spectrum with weights for the \(p\)-Laplacian.

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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