Abstract
In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem \begin{equation*}a(u, v) = \lambda b(u, v) + \mu m(u, v) + \varepsilon F(u, v),\end{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.
Citation
Daniel Maroncelli. Mauricio A. Rivas. "A Fredholm alternative for elliptic equations with interior and boundary nonlinear reactions." Topol. Methods Nonlinear Anal. 62 (1) 135 - 157, 2023. https://doi.org/10.12775/TMNA.2022.054
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