PDE Seminar Abstracts

Zhaoli Liu

Capital Normal University, Beijing

10 Aug 2009 3-4pm, Carslaw Room 454

Capital Normal University, Beijing

10 Aug 2009 3-4pm, Carslaw Room 454

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 $f\left(x,t\right)\u2215t$ crosses several eigenvalues of $-\Delta $. Similar results are proved for (S) as well as for Hamiltonian systems. Here we do not need $f\left(x,t\right)\u2215t$ 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.)

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