Bubbles clustered inside for almost critical problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912221296066560 |
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| author | Ayed, Mohamed Ben Mehdi, Khalil El |
| author_facet | Ayed, Mohamed Ben Mehdi, Khalil El |
| contents | We investigate the existence of blowing-up solutions of the following almost critical problem $$ -Δu +V(x)u =u^{p-\e},\quad u>0\quad\mbox{in}\quad Ø,\quad u=0\quad\mbox{on}\quad \partialØ, $$
where $Ø$ is a bounded regular domain in $\mathbb{R}^n$, $n\geq 4$, $\varepsilon$ is a small positive parameter, $p+1=(2n)/(n-2)$ is the critical Soblolev exponent and the potential $V$ is a smooth positive function. We find solutions which exhibit bubbles clustered inside as $\e$ goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_03235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bubbles clustered inside for almost critical problems Ayed, Mohamed Ben Mehdi, Khalil El Analysis of PDEs 35A15, 35J20, 35J25 We investigate the existence of blowing-up solutions of the following almost critical problem $$ -Δu +V(x)u =u^{p-\e},\quad u>0\quad\mbox{in}\quad Ø,\quad u=0\quad\mbox{on}\quad \partialØ, $$ where $Ø$ is a bounded regular domain in $\mathbb{R}^n$, $n\geq 4$, $\varepsilon$ is a small positive parameter, $p+1=(2n)/(n-2)$ is the critical Soblolev exponent and the potential $V$ is a smooth positive function. We find solutions which exhibit bubbles clustered inside as $\e$ goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional. |
| title | Bubbles clustered inside for almost critical problems |
| topic | Analysis of PDEs 35A15, 35J20, 35J25 |
| url | https://arxiv.org/abs/2502.03235 |