A general theory for the $(s, p)$-superposition of nonlinear fractional operators
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arXiv
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| Format: | Preprint |
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2024
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| author | Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico |
| author_facet | Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico |
| contents | We consider the continuous superposition of operators of the form \[ \iint_{[0, 1]\times (1, N)} (-Δ)_p^s \,u\,dμ(s,p), \] where $μ$ denotes a signed measure over the set $[0, 1]\times (1, N)$, joined to a nonlinearity satisfying a proper subcritical growth. The novelty of the paper relies in the fact that, differently from the existing literature, the superposition occurs in both $s$ and $p$.
Here we introduce a new framework which is so broad to include, for example, the scenarios of the finite sum of different (in both $s$ and $p$) Laplacians, or of a fractional $p$-Laplacian plus a $p$-Laplacian, or even combinations involving some fractional Laplacians with the "wrong" sign.
The development of this new setting comes with two applications, which are related to the Weierstrass Theorem and a Mountain Pass technique. The results obtained contribute to the existing literature with several specific cases of interest which are entirely new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A general theory for the $(s, p)$-superposition of nonlinear fractional operators Dipierro, Serena Lippi, Edoardo Proietti Sportelli, Caterina Valdinoci, Enrico Analysis of PDEs We consider the continuous superposition of operators of the form \[ \iint_{[0, 1]\times (1, N)} (-Δ)_p^s \,u\,dμ(s,p), \] where $μ$ denotes a signed measure over the set $[0, 1]\times (1, N)$, joined to a nonlinearity satisfying a proper subcritical growth. The novelty of the paper relies in the fact that, differently from the existing literature, the superposition occurs in both $s$ and $p$. Here we introduce a new framework which is so broad to include, for example, the scenarios of the finite sum of different (in both $s$ and $p$) Laplacians, or of a fractional $p$-Laplacian plus a $p$-Laplacian, or even combinations involving some fractional Laplacians with the "wrong" sign. The development of this new setting comes with two applications, which are related to the Weierstrass Theorem and a Mountain Pass technique. The results obtained contribute to the existing literature with several specific cases of interest which are entirely new. |
| title | A general theory for the $(s, p)$-superposition of nonlinear fractional operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.14049 |