Existence results for resonant perturbations of the Fučik spectrum
Let \(\Omega\) be a bounded smooth domain in \(\mathbb{R}^N\) and consider the problem \[ -\Delta u=\lambda u^+- \alpha u^-\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega,\tag{\(*\)} \] where \(u^+= \max(u, 0)\) and \(u= u^+- u^-\). The set \(\Sigma_0= \{(\alpha, \lambda)\in \mathbb{R}^2\): there exists \(u\in H^1_0(\Omega)\), \(u\neq 0\) solution of \((*)\}\) is called the Fučik spectrum and has been studied by many authors during the last years. Here, the authors study a nonlinear perturbations to the above problem by adding a subcritical nonlinearity \(g(x,u)\) to the right-hand side where \((\alpha,\lambda)\in C_2\), \(C_2\) being a continuous curve having certain properties contained in the Fučik spectrum. The problem is resonant in the sense that \(2G(x, s)/s^2\) tends to zero at infinity uniformly in \(x\), where \(G\) is the primitive of \(g\). Two non-quadraticity conditions at infinity \((NQ)_{\pm}\) are also considered. The main result (Theorem 1) asserts that if \((NQ)_+\) and the resonance condition are satisfied there is (at least) a weak solution. The proof uses a variant of the Saddle-Point theorem by Rabinowitz. The case of periodic boundary conditions for \(N= 1\) is also studied by using the variational characterization in this case of the Fučik spectrum, a version of the Palais-Smale condition due to G. Cerami and a higher linking variational argument.
scientific article; zbMATH DE number 1127700