Construction of solutions of an supercritical elliptic PDE in low dimensions

Abstract

In this paper, we study the supercritical problem \((P_\varepsilon )\): \( -\triangle u=K|u|^{4/(n-2)+\varepsilon }u~ \text{ in } \Omega , ~ u=0 \text{ on } \partial \Omega ,\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^n\), \(n=3,4\), \(\varepsilon \) is a positive real parameter and K is a \(C^4\) positive function on \({\overline{\Omega }}\). Following the ideas of Bahri et al. (Calc Var Partial Differ Equ 8:67–93, 1995) and using the finite-dimensional reduction, we construct some solutions of \((P_\varepsilon )\).

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Correspondence to Yessine Dammak.

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Ben Ayed, M., Dammak, Y. Construction of solutions of an supercritical elliptic PDE in low dimensions. Annali di Matematica 200, 51–66 (2021). https://doi.org/10.1007/s10231-020-00982-7

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Keywords

  • Supercritical exponent
  • Finite-dimensional reduction
  • Critical points

Mathematics Subject Classification

  • Primary 35J20
  • Secondary 35J60