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

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) = \alpha m(x) u^+ \beta u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} \sigma (x) =\alpha \rho u^+- +g(x,u), \partial }$$ where \((\alpha, \beta) \mathbb{R}^2\), \(c, m L^\infty (\Omega)\), \(\sigma, (\partial\Omega)\), and nonlinearities f g are bounded continuous functions. study asymmetric (Fucik) spectrum with weights, prove existence theorems for nonlinear perturbations of this both resonance non-resonance cases. For case, we provide a sufficient condition, so-called generalized Landesman-Lazer solvability. The proofs based on variational methods rely strongly characterization spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html