Nonlinear elliptic equations and systems with a kind of twisted nonlinearities

Abstract

Consider the elliptic equation (E) $-\Delta u=f\left(x,u\right)$ in $\Omega$ with $u=0$ on $\partial \Omega$ and the elliptic system (S) $-\Delta u={\nabla }_{u}V\left(x,u\right)$ in $\Omega$ subject to $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $R^N$ with smooth boundary $\partial \Omega$. Under suitable conditions on $f:\Omega \times R\to R$ and $V:\Omega \times R^m\to R$, nontrivial solutions are obtained for (E) provided that $f\left(x,t\right)∕t$ crosses several eigenvalues of $-\Delta$. Similar results are proved for (S) as well as for Hamiltonian systems. Here we do not need $f\left(x,t\right)∕t$ to have an asymptotic limit, which was assumed in the literature for similar problems. (This is joint work with Jiabao Su and Zhi-Qiang Wang.)