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Bubbles clustered inside for almost critical problems

Mohamed Ben Ayed, Khalil El Mehdi

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.

Bubbles clustered inside for almost critical problems

Abstract

We investigate the existence of blowing-up solutions of the following almost critical problem where is a bounded regular domain in , , is a small positive parameter, is the critical Soblolev exponent and the potential is a smooth positive function. We find solutions which exhibit bubbles clustered inside as 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.

Paper Structure

This paper contains 6 sections, 14 theorems, 109 equations.

Table of Contents

  1. Introduction and Main Results
  2. Analytical Framework
  3. Estimate of the infinite-dimensional part
  4. Expansion of the gradient of the associated functional
  5. Interior blowing up solutions with clustered bubbles
  6. Appendix

Key Result

Theorem 1.1

Let $n \geq 4$, $N \geq 2$ and $b \in \Omega$ be a non-degenerate critical point of $V$. Assume that the function $\mathcal{F}_{ b , N }$ has a non-degenerate critical point $( \overline{z}_1, \cdots, \overline{z}_N)$. Then, there exists a small positive real $\varepsilon _0$ such that for any $\va with where $\eta_0$ is a small positive constant, $\sigma_4=1$, $\sigma_n=0$ for $n\geq 5$, $c$ is

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 3.1
  • Lemma 3.2
  • ...and 5 more