Nondegeneracy of elliptic problems with periodic nonlinearities in the plane
Let \(\Omega \subset\mathbb R^N\) be a bounded domain with a smooth boundary and let \(\mathcal{S}_0 = \{ \widetilde{h} \in L^2(\Omega): \int_\Omega \widetilde{h} (\omega)\varphi(\Omega) \,d\omega = 0\}\), with \(\varphi\) being a generator of the one dimensional eigenspace of the first eigenvalue \(\mu_1\) of the Laplace operator with homogeneous Dirichlet boundary conditions. The author investigates qualitative results of existence of solutions for the following semilinear, strongly resonant, elliptic boundary value problem \[ -\Delta u - \mu_1 u + g(u) = h \quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial \Omega, \tag{P} \] where the \(C^1\) nonlinearity \(g\) is assumed to be nontrivial, \(2\pi\)-periodic and to have zero mean value. Denote by \(S_g\) the set of the \(\widetilde{h} \in L^2(\Omega)\) such that the problem (P) is solvable. The A. proves the following result: Let \(N=2\); then the set \(S_g\) contains the hyperplane \(S_0\) into its \(L(\Omega)\)-interior. The result is an extension of a previous result by \textit{S. Solimini} [J. Math. Anal. Appl. 117, 138--152 (1986; Zbl 0634.35030)] to the planar case.
scientific article; zbMATH DE number 5164772