An existence theory for nonlinear superposition operators of mixed fractional order
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914385136451584 |
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| author | Dipierro, Serena Perera, Kanishka Sportelli, Caterina Valdinoci, Enrico |
| author_facet | Dipierro, Serena Perera, Kanishka Sportelli, Caterina Valdinoci, Enrico |
| contents | We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02628 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An existence theory for nonlinear superposition operators of mixed fractional order Dipierro, Serena Perera, Kanishka Sportelli, Caterina Valdinoci, Enrico Analysis of PDEs We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure. |
| title | An existence theory for nonlinear superposition operators of mixed fractional order |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.02628 |