Multiple solutions of semilinear Dirichlet problems on a domain with \(S^1\)-group actions
The author considers the Dirichlet problem for the non-homogeneous semilinear elliptic equation \[ \Delta u + g(u) = h, \tag{P} \] on \(S^1\)-symmetric domains in \(\mathbb R^N\). The author supposes that the nonlinearity always crosses at least 3 eigenvalues of the corresponding linear boundary value problem. When \(\| h\| _{L^2}\) is small and \(S^1\)-invariant, the existence of at least \(3\) solutions is proved. When \(\| h\| _{L^2}\) is small and not \(S^1\)-invariant, the existence of at least \(5\) solutions is proved. In the proof, the author uses Morse theory and topological degree methods. This result is motivated by results of \textit{A. Ambrosetti} and \textit{G. Mancini} [Nonlinear Anal., Theory Methods Appl. 3, 635-645 (1979; Zbl 0433.35025)], and \textit{L. Boccardo} and \textit{T. Gallouët} [Nonlinear Anal., Theory Methods Appl. 14, No. 1, 81-91 (1990; Zbl 0713.35029)], who proved the existence of exactly \(3\) solutions when the nonlinearity crosses precisely one corresponding linear eigenvalue. This problem is related to a general conjecture on the relations of the number of linear eigenvalues crossed by the nonlinearity and the number of solutions of the corresponding nonlinear problem.
scientific article; zbMATH DE number 1123524
scientific article