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Existence of solutions to a superlinear \(p\)-Laplacian equation. (English) Zbl 1011.35062
Existence of nontrivial solutions for the Dirichlet problem \(-\Delta_p u= f(x,u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) is studied. Here, \(-\Delta_p u\) is the \(p\)-Laplacian, \(p>1\), and \(f\) is a Caratheodory function, which has “superlinear” and subcritical growth for large \(|u|\). For small \(|u|\), \(f\) is assumed to behave like \(\lambda|u|^{p-2}u\), where \(\lambda\) is between the first and the second Dirichlet eigenvalue of the \(p\)-Laplacian. The corresponding variational functional is studied by means of Morse theory.

35J65 Nonlinear boundary value problems for linear elliptic equations
35B34 Resonance in context of PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
49J35 Existence of solutions for minimax problems
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