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On a critical point theorem and an application to a nonresonance problem between two eigenvalues of the \(p\)-Laplacian. (Sur un théorème de point critique et application à un problème de non-résonance entre deux valeurs propres du \(p\)-laplacien.) (French. English summary) Zbl 0971.35031
Summary: We establish an abstract theorem of critical point theory which constitutes a generalization of classical link theorems. We also establish some variational characterizations of the \(p\)-Laplacian spectrum. As application we prove the existence of solutions of the problem \[ -\Delta_p u= f(x,u)\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega, \] where the ratios \({f(x,s)\over|s|^{p- 2}s}\) and \({pF(x,s)\over|s|^p}\) lie between two consecutive eigenvalues of the \(p\)-Laplacian and \(F\) is a potential of \(f\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35P05 General topics in linear spectral theory for PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35A15 Variational methods applied to PDEs
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