Multiple Existence

Auszug

In this chapter we study the existence of the solution for

$$ \begin{gathered} - \Delta \upsilon + a\upsilon = \frac{{\lambda e^\upsilon }} {{\smallint _\Omega e^\upsilon dx}}in \Omega , \hfill \\ \frac{{\partial \upsilon }} {{\partial \nu }} = 0on\partial \Omega , \hfill \\ \end{gathered} $$

(8.1)

where Ω ⊂ R² is a bounded domain with smooth boundary ∂Ω. It is the stationary problem of (3.1), that is, (7.15) with W(x = 1, and we will obtain several suggestions to the nonstationary problem. In this special case of W (x, problem (8.1) admits a constant solution, and this trivial solution generates nontrivial ones.