Abstract multiplicity theorems and applications to critical growth problems
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917756955262976 |
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| author | Perera, Kanishka |
| author_facet | Perera, Kanishka |
| contents | We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian and $(p,q)$-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $λ> 0$. In particular, the number of solutions goes to infinity as $λ\to \infty$. Moreover, we give an explicit lower bound on $λ$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues constructed using the ${\mathbb Z}_2$-cohomological index. This is a consequence of the fact that our abstract multiplicity results make essential use of the piercing property of the cohomological index, which is not shared by the genus. Our result for the $p$-Laplacian is new even in the semilinear case $p = 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07901 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Abstract multiplicity theorems and applications to critical growth problems Perera, Kanishka Analysis of PDEs Primary 58E05, Secondary 35J92, 35B33 We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian and $(p,q)$-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $λ> 0$. In particular, the number of solutions goes to infinity as $λ\to \infty$. Moreover, we give an explicit lower bound on $λ$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues constructed using the ${\mathbb Z}_2$-cohomological index. This is a consequence of the fact that our abstract multiplicity results make essential use of the piercing property of the cohomological index, which is not shared by the genus. Our result for the $p$-Laplacian is new even in the semilinear case $p = 2$. |
| title | Abstract multiplicity theorems and applications to critical growth problems |
| topic | Analysis of PDEs Primary 58E05, Secondary 35J92, 35B33 |
| url | https://arxiv.org/abs/2308.07901 |