Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913481387671552 |
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| author | Boscaggin, Alberto Colasuonno, Francesca Noris, Benedetta Weth, Tobias |
| author_facet | Boscaggin, Alberto Colasuonno, Francesca Noris, Benedetta Weth, Tobias |
| contents | We deal with the following semilinear equation in exterior domains \[-Δu + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03029 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains Boscaggin, Alberto Colasuonno, Francesca Noris, Benedetta Weth, Tobias Analysis of PDEs 35J20, 35B06, 35B09 We deal with the following semilinear equation in exterior domains \[-Δu + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. 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. |
| title | Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains |
| topic | Analysis of PDEs 35J20, 35B06, 35B09 |
| url | https://arxiv.org/abs/2309.03029 |