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Asymmetric elliptic problems with indefinite weights. (English) Zbl 1016.35054

The present paper studies the following quasilinear eigenvalue problem \[ \begin{cases} -\Delta_p u=\lambda[m(x)(u^+)^{p-1}- n(x)(u^-)^{p-1}\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(u^{\pm}:= \max(\pm u,0)\), \(m(x)\) and \(n(x)\) are given functions, and \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). The authors prove the existence of a first nontrivial eigenvalue of (1). Applications of the obtained results to the description of the Fučik spectrum and to the study of the corresponding nonresonance problems are also given.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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