PDE Seminar: Liu -- Nonlinear elliptic equations and systems with a kind of twisted nonlinearities

 Speaker: Zhaoli Liu from Capital Normal University, Beijing 

3:00pm-4:00pm, Monday 10th August, Carslaw 454 

Title: Nonlinear elliptic equations and systems with a kind of twisted nonlinearities 

Abstract: Consider the elliptic equation (E) $ -\Delta u = f(x, u)$ in $\Omega$ with
$u=0$ on $\partial \Omega$ and the elliptic system (S) $-\Delta u = \nabla_u V(x, u)$ 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 $f(x,t)/t$ crosses several eigenvalues of $-\Delta$.  Similar
results are proved for (S) as well as for Hamiltonian systems.  Here we do not need
$f(x,t)/t$ 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.)