The jumping nonlinearity problem revisited: an abstract approach

Abstract

We consider a class of nonlinear problems of the form $$ Lu+g(x,u)=f, $$ where $L$ is an unbounded self-adjoint operator on a Hilbert space $H$ of $L^{2}(\Omega)$-functions, $\Omega\subset\mathbb{R}^{N}$ an arbitrary domain, and $g\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a "jumping nonlinearity" in the sense that the limits $$ \lim_{s\rightarrow-\infty} \frac{g(x,s)}{s}=a \quad\text{and}\quad \lim_{s\rightarrow\infty}\frac{g(x,s)}{s}=b $$ exist and "jump" over an eigenvalue of the operator $-L$. Under rather general conditions on the operator $L$ and for suitable $a< b$, we show that a solution to our problem exists for any $f\in H$. Applications are given to the beam equation, the wave equation, and elliptic equations in the whole space $\mathbb{R}^{N}$.