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

Abstract

We study the existence of solutions of the Dirichlet problem {gather} -ϕ_p(u')' -a_+ ϕ_p(u^+) + a_- ϕ_p(u^-) -λϕ_p(u) = f(x,u), \quad x \in (0,1), \label{pb.eq} \tag{1} u(0)=u(1)=0,\label{pb_bc.eq} \tag{2} {gather} where $p>1$, $ϕ_p(s):=|s|^{p-1}\sgn s$ for $s \in \mathbb{R}$, the coefficients $a_\pm \in C^0[0,1]$, $λ\in \mathbb{R}$, and $u^\pm := \max\{\pm u,0\}$. We suppose that $f\in C^1([0,1]\times\mathbb{R})$ and that there exists $f_\pm \in C^0[0,1]$ such that $\lim_{ξ\to\pm\infty} f(x,ξ) = f_\pm(x)$, for all $x \in [0,1]$. With these conditions the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to have a `jumping nonlinearity'. We also suppose that the problem {gather} -ϕ_p(u')' = a_+ ϕ_p(u^+) - a_- ϕ_p(u^-) + λϕ_p(u) \quad\text{on} \ (0,1), \tag{3} \label{heval_pb.eq} {gather} together with \eqref{pb_bc.eq}, has a non-trivial solution $u$. That is, $λ$ is a `half-eigenvalue' of \eqref{pb_bc.eq}-\eqref{heval_pb.eq}, and the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem \eqref{pb.eq}-\eqref{pb_bc.eq} has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients $a_\pm$ 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.