## Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus.(English)Zbl 0803.35053

From the introduction: We study the existence of many positive nonradial solutions of the equation $\Delta u + f(u) = 0 \quad \text{in} \quad \Omega, \qquad u = 0 \quad \text{on} \quad \partial \Omega,$ where $$\Omega = \Omega_ a = \{x \in \mathbb{R}^ n : a < | x |<1\}$$ is an annulus in $$\mathbb{R}^ n$$, $$n \geq 2$$, and $$f$$ satisfies the following conditions: $$f \in C^ 1 (\mathbb{R}^ 1)$$ and $$f(u)>0$$ for $$u$$ large, $$f(0)=0$$ and $$f'(0) \leq 0$$, there exists $$\sigma>0$$ such that $$uf'(u) \geq (1+ \sigma) f(u)$$ for all $$u \geq 0$$, for $$u$$ large, $f(u) \leq \begin{cases} Cu^ p \quad & \text{ for some } \;p<(n + 2)/(n-2) \quad \text{ and } \quad C>0 \text{ if } n \geq 3,\\ \exp A(u) & \text{ with } \;A(u)=o(u^ 2) \quad \text{ as } \quad u \to \infty \quad \text{ if } \quad n = 2.\end{cases}$ {}.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 58J70 Invariance and symmetry properties for PDEs on manifolds 35J20 Variational methods for second-order elliptic equations
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