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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.

MSC:
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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