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Remarks on multiple nontrivial solutions for quasi-linear resonant problems. (English) Zbl 1050.35025
Summary: In this paper Morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of Dirichlet boundary value problems with double resonance at infinity and at 0, viz. $-\Delta_p(u)=f(x,u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial\Omega,$ where $$\Omega\subset\mathbb R^N$$ is a bounded smooth domain, $$\Delta_p(u)=\text{div}(|\nabla u|^{p-2}\nabla u)$$ is the $$p$$-Laplacian with $$1<p<N$$ and $$f(x,u)=O(u^{p-1})$$ as $$u\to 0$$ and $$u\to\infty$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J60 Nonlinear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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