Asymptotically linear systems near and at resonance
Maya Chhetri, Petr Girg · Boundary Value Problems · 2014
This paper deals with an elliptic system of the formu = λθ 1 a(x) is a bounded domain with C 2,ξ -boundary ∂ , ξ ∈ (0, 1) (a bounded open interval if N = 1).Here a(x) ∈ L ∞ ( ) with a(x) > 0 a.e. in and θ 1 , θ 2 > 0 are constants.The nonlinear perturbations f , g : R × × R → R are Carathéodory functions that are sublinear at infinity.We provide sufficient conditions for determining the λ-direction to which a continuum of positive and negative solutions emanates from infinity at the first eigenvalue of the associated linear problem.Furthermore, as a consequence of main results, we also provide sufficient condition for the solvability of a class of asymptotically linear system near and at resonance satisfying Landesman-Lazer type conditions.