Type (II) regions between curves of the Fučik spectrum (Q1376804)

The Fucik spectrum arises in the study of semilinear elliptic boundary value problems of the form (1) \(Au=f(x,u)\), where \(A\) is a selfadjoint operator having compact resolvent on \(L^2(\Omega)\), \(\Omega\subset \mathbb{R}^n\), and \(f(x,t)\) is a Carathéodory function on \(\overline\Omega \times\mathbb{R}\) such that \(f(x,t)/t\to a\) a.e. as \(t\to-\infty\), \(f(x,t)/t\to b\) a.e. as \(t\to+\infty\). The author considers the situation when \((a,b)\) is in the region between the curves \(C_{l_1}, C_{l_2}\) (when they do not coincide) and is not in \(\Sigma\) (the Fučik spectrum of \(A)\). In this case, if \[ f(x,t)=bt^+ -at^-+p(x,t), \quad \text{where } \bigl| p(x,t) \bigr|\leq C\bigl(| t|+ 1\bigr), \] and the subspace of eigenfunctions corresponding to \(\lambda_l\) is contained in \(L^\infty (\Omega)\), the author proves that (1) has a nontrivial solution. This result holds even if \(\lambda_l\) is a multiple eigenvalue. In the proof of the main result, the author uses new linking methods in saddle point theory.