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

We consider the boundary value problem -Delta u + c(x)u = alpha m(x)u(+) - beta m(x)u(-) + f(x, u), x is an element of Omega, partial derivative u/partial derivative eta + sigma(x)u = alpha rho(x)u(+) - beta rho(x)u(-) + g(x, u), x is an element of partial derivative Omega, where (alpha, beta) is an element of R-2, c, m is an element of L-infinity(Omega), sigma, rho is an element of L-infinity (partial derivative 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.