Pairs of nontrivial smooth solutions for nonlinear Neumann problems (Q2184943)

The author studies a nonlinear Neumann problem associated to a divergence form equation where the function \(a(\nabla u(z))\), involved in the highest order derivatives, is a strictly monotone and continuous map which satisfies certain regularity and growth properties and the reaction term \(f(z, u(z))\) exhibits strong resonance at infinity. These conditions provide a general framework in which it is possible to fit many differential operators of interest such as the \(p\)-Laplacian and the \((p,q)\)-Laplacian which appears in many models of physical processes. Arguing by contradiction and using Bolzano's theorem and variational instruments based on critical point theory, the author proves the existence of two nontrivial smooth solutions.