A Fredholm alternative for elliptic equations with interior and boundary nonlinear reactions - Maroncelli, Daniel; Rivas, Mauricio A. (Topological Methods in Nonlinear Analysis, 2023)
In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem equation* a(u, v) = λ b(u, v) + μ m(u, v) + ε F(u, v), equation* for a triple (a, b, m) of continuous, symmetric bilinear forms on a real separable Hilbert space V and nonlinear form F. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde’s with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results.