Bubbles clustered inside for almost critical problems

Mohamed Ben Ayed; Khalil El Mehdi

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arXiv · 2502.03235

Bubbles clustered inside for almost critical problems

Mohamed Ben Ayed·Khalil El Mehdi

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Abstract

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.

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Mohamed Ben Ayed, Khalil El Mehdi. 2025-02-05. Bubbles clustered inside for almost critical problems. https://arxiv.org/abs/2502.03235

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