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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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