Multiple sign-changing solutions to semilinear elliptic resonant problems. (English) Zbl 1185.35096

Summary: We use the topological degree theory and the critical groups to investigate the elliptic equation \(-\Delta u = f(x,u)\) in \(\Omega \) and subject to \(u = 0\) on \(\partial \Omega \), and establish a multiple solutions theorem which guarantees that this problem has at least six nontrivial solutions under some resonant conditions. If this problem has only finitely many solutions then, among them, there are two positive solutions, two negative solutions and two sign-changing solutions.


35J61 Semilinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B09 Positive solutions to PDEs
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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