Landesman–Lazer conditions for resonant p-Laplacian problems with jumping nonlinearities

We study the existence of solutions of the p-Laplacian Dirichlet problemequation(1)−ϕp(u′)′=λϕp(u)+h(x,u)+f(x,u,u′),x∈(0,1),equation(2)u(0)=u(1)=0,u(0)=u(1)=0, where λ∈Rλ∈R, p>1p>1, ϕs(ξ):=|ξ|s−1sgnξ for s⩾1s⩾1, ξ∈Rξ∈R, the function h:[0,1]×R→Rh:[0,1]×R→R has the formh(x,ξ)=a+∞(x)ϕp(ξ+)−a−∞(x)ϕp(ξ−),(x,ξ)∈[0,1]×R, with ξ±:=max⁡{±ξ,0}ξ±:=max⁡{±ξ,0}, and a±∞∈L1(0,1)a±∞∈L1(0,1), and the function f:[0,1]×R2→Rf:[0,1]×R2→R is continuous, and satisfies|f(x,ξ,η)|⩽K(x)(1+|ξ|q−1),(x,ξ,η)∈[0,1]×R2, for some q∈[1,p)q∈[1,p) and K∈L1(0,1)K∈L1(0,1).The dominant asymptotic behaviour of equation (1) as u→±∞u→±∞ is determined by the coefficients a±∞a±∞, and we allow a−≠a+a−≠a+, in which case the problem is said to be jumping  . If the positively homogeneous problem obtained from (1)–(2) by setting f≡0f≡0 has a non-trivial solution then the problem is said to be resonant, and λ is said to be a half-eigenvalue. Assuming that the problem (1)–(2) is both jumping and resonant, we will obtain a solution under certain ‘Landesman–Lazer’ conditions on f.