Special Issue in honor of John W. Neuberger. Electron. J. Diff. Eqns., Special Issue 02 (2023), pp. 209-230.

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

Nsoki Mavinga, Quinn A. Morris, Stephen B. Robinson

Abstract:
We consider the boundary value problem

where $(\alpha, \beta) \in \mathbb{R}^2$, $c, m \in L^\infty (\Omega)$, $\sigma, \rho \in L^\infty (\partial\Omega)$, and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum.

Published March 27, 2023.
Math Subject Classifications: 35P30, 35J60, 35J66.
Key Words: Fucik Spectrum; resonance; nonlinear boundary condition.
DOI: https://doi.org/10.58997/ejde.sp.02.m2

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Nsoki Mavinga
Department of Mathematics & Statistics
Swarthmore College
Swarthmore, PA 19081-1390, USA
email: nmaving1@swarthmore.edu
Quinn A. Morris
Department of Mathematical Sciences
Appalachian State University
Boone, NC 28608, USA
email: morrisqa@appstate.edu
Stephen B. Robinson
Department of Mathematics
Wake Forest University
Winston-Salem, NC 27109, USA
email: sbr@wfu.edu

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