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Nonlinear elliptic equations on expanding symmetric domains. (English) Zbl 0944.35026

The authors consider the problem of the existence of nonradial solutions for the semilinear elliptic equation \[ -\Delta u + u = u^p , \;u > 0, \quad \text{in } \Omega_a ,\qquad u = 0 \quad \text{on }\partial \Omega_a \] on expanding (as \(a \rightarrow \infty\)) annulus \[ \Omega_a = \{x \in {\mathbb R}^n : a < |x|< a + 1 \}. \] They introduce a new approach and show how to construct solutions that are obtained as local minimizers of the energy. Their method enables them to get some qualitative properties of solutions such as the shape and the exact symmetry of solutions. For instance, they prove that all solutions obtained are multi-bump solutions with a discrete number of bumps. Under certain conditions they can construct nonradial solutions with prescribed symmetry. They also indicate how the results can be extended to more general domains and more general equations.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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