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

Original · EN

We study the existence of solutions of the Dirichlet problem gather -ϕₚ(u')' -a+ ϕₚ(u+) + a- ϕₚ(u-) -λϕₚ(u) = f(x,u), x ∈ (0,1), pb.eq 1 u(0)=u(1)=0,pbbc.eq 2 gather where p>1, ϕₚ(s):=|s|ᵖ⁻¹ s for s ∈ R, the coefficients a± ∈ C⁰[0,1], λ∈ R, and u±:= {± u,0}. We suppose that f∈ C¹([0,1]) and that there exists f± ∈ C⁰[0,1] such that ξ→±∞ f(x,ξ) = f±(x), for all x ∈ [0,1]. With these conditions the problem pb.eq-pbbc.eq is said to have a `jumping nonlinearity'. We also suppose that the problem gather -ϕₚ(u')' = a+ ϕₚ(u+) - a- ϕₚ(u-) + λϕₚ(u) (0,1), 3 hevalₚb.eq gather together with pbbc.eq, has a non-trivial solution u. That is, λ is a `half-eigenvalue' of pbbc.eq-hevalₚb.eq, and the problem pb.eq-pbbc.eq is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem pb.eq-pbbc.eq 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 `Fucik 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 Fucik spectrum.