Landesman–Lazer conditions at half-eigenvalues of the p-Laplacian

We study the existence of solutions of the Dirichlet problemequation(1)−ϕp(u′)′−a+ϕp(u+)+a−ϕp(u−)−λϕp(u)=f(x,u),x∈(0,1),equation(2)u(0)=u(1)=0,u(0)=u(1)=0, where p>1p>1, ϕp(s):=|s|p−1sgns for s∈Rs∈R, the coefficients a±∈C0[0,1]a±∈C0[0,1], λ∈Rλ∈R, and u±:=max{±u,0}u±:=max{±u,0}. We suppose that f∈C1([0,1]×R)f∈C1([0,1]×R) and that there exist f±∈C0[0,1]f±∈C0[0,1] such that limξ→±∞f(x,ξ)=f±(x), for all x∈[0,1]x∈[0,1]. With these conditions the problem  and  is said to have a ‘jumping nonlinearity’. We also suppose that the problemequation(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  and , and the problem  and  is said to be ‘resonant’. Combining a shooting method with so-called ‘Landesman–Lazer’ conditions, we show that the problem  and  has a solution.Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients a±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.