Remarks on multiple nontrivial solutions for quasi-linear resonant problems. (English) Zbl 1050.35025

Liu, Jiaquan; Su, Jiabao

J. Math. Anal. Appl. 258, No. 1, 209-222 (2001).

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

Keywords:

Quasi-linear elliptic equation; double resonance; critical group; homological nontrivial critical point; Morse theory; local linking; multiple solutions

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