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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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