Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains. (English) Zbl 07916676
Summary: We deal with the following semilinear equation in exterior domains
\[
-\Delta u+u=a(x)|u|^{p-2}u,\quad u\in H_0^1(A_R),
\]
where \(A_R:=\{x\in\mathbb{R}^N:|x|>R\}\), \(N\geqslant 3\), \(R>0\). Assuming that the weight \(a\) is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as \(a\), for values of \(p>2\) in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight \(a\), our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical \(p\) in nonradial exterior domains.
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{© 2024 IOP Publishing Ltd & London Mathematical Society. All rights, including for text and data mining, AI training, and similar technologies, are reserved.}
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35J15 | Second-order elliptic equations |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35B09 | Positive solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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