PDE Seminar Abstracts

Lin and Ni conjectured that the Neuman system

$$\begin{array}{ccccc}\hfill -\Delta u+\epsilon u& ={u}^{\frac{n+2}{n-2}}\hfill & \hfill & \text{in}\Omega \subset \subset {\mathbb{R}}^{N}\text{}\hfill & \hfill \\ \hfill u& 0\hfill & \hfill & \text{in}\Omega \text{}\hfill \\ \hfill {\partial}_{\nu}u& =0\hfill & \hfill & \text{in}\partial \Omega \text{}\hfill \end{array}$$

admits only the constant solution for small $\epsilon >0$. This conjecture has stimulated and generated intensive contributions in the past decades. In particular, it has been known for long that the conjecture is not valid in small dimensions (except 3). Moreover, recent exemples of Wei show that it is not valid when the mean curvature is negative and for innite energies. We prove here that the conjecture is valid in the positive mean curvature case and with nite energy when $n=3$ or $n\ge 7$. This is joint work with O.Druet and J.Wei.

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