Multiplicity of solutions for doubly resonant Neumann problems (Q2430159)

A semilinear elliptic Neumann problem of the form \(-\Delta x(z)=f(z,x(z))\) is studied on a \(C^2\) bounded domain \(Z\subset\mathbb R^N\). The right-hand side \(f(x,z)\) is supposed to be of the form \(f(z,x)=\lambda_k x+g(z,x)\) with \(\lim_{|x|\to\infty}g(z,x)/x=0\) uniformly in \(z\). Here \(\lambda_k\) is assumed to be a Neumann eigenvalue of \(-\Delta\) on \(Z\), in which case the problem is said to be resonant at infinity with respect to \(\lambda_k\). It is called doubly resonant, if a similar condition also holds for two consecutive eigenvalues \(\lambda_k< \lambda_{k+1}\). Doubly resonant problems for the Dirichlet problem have been studied, starting with [\textit{H. Berestycki} and \textit{D. Guedes de Figueiredo}, Commun.\ Partial Differ.\ Equations 6, 91--120 (1981; Zbl 0468.35043)]. Two rather technical sets of conditions on the nonlinearity \(f\) are described here that imply the existence of multiple (two or three) solutions. The variational proofs involve Morse theory.