Nonlinear Dirichlet problems with a crossing reaction
We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplacian (p > 2) and a Laplacian, with a reaction that is (p-1)-sublinear and exhibits an asymmetric behavior near ∞ and -∞, crossing ^λ1 > 0, the principal eigenvalue of (-Δp,W01,p(Ω)) (crossing nonlinearity). Resonance with respect to ^λ1(p) > 0 can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction C1 in the x-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.
nonlinear regularity, critical groups., resonance, nonlinear maximum principle
Hu, Shouchuan, and Nikolaos S. Papageorgiou. "Nonlinear Dirichlet problems with a crossing reaction." Communications on Pure & Applied Analysis 13, no. 6 (2014): 2749.
Communications on Pure & Applied Analysis