2 LANDESMAN-LAZER CONDITIONS AT HALF-EIGENVALUES OF THE p-LAPLACIAN
Bryan Patrick Rynne · 2016
We study the existence of solutions of the Dirichlet problem(1)−ϕp(u′)′−a+ϕp(u+)+a−ϕp(u−)−λϕp(u)=f(x,u),x∈(0,1),(2)u(0)=u(1)=0, where p>1, ϕp(s):=|s|p−1sgns for s∈R, the coefficients a±∈C0[0,1], λ∈R, and u±:=max{±u,0}. We suppose that f∈C1([0,1]×R) and that there exist f±∈C0[0,1] such that limξ→±∞f(x,ξ)=f±(x), for all x∈[0,1]. With these conditions the problem (1), (2) is said to have a ‘jumping nonlinearity’. We also suppose that the problem(3)−ϕp(u′)′=a+ϕp(u+)−a−ϕp(u−)+λϕp(u)on (0,1), together with (2), has a non-trivial solution u. That is, λ is a ‘half-eigenvalue’ of (2), (3), and the problem (1), (2) is said to be ‘resonant’. Combining a shooting method with so-called ‘Landesman–Lazer’ conditions, we show that the problem (1), (2) has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients a± are constants, and the resonance has been at a point in the ‘Fučík spectrum’. Even in this constant coefficient case our result extends previous results. In particular, previous variational approaches have required strong conditions on the location of the resonant point, whereas our result applies to any point in the Fučík spectrum.