Existence and uniqueness results on Single-Peaked solutions of a semilinear problem

  • Daomin Cao

    Young Scientist laboratory of Mathematical Physics. Wuhan Institute of Mathematical Sciences, The Chinese Academy of Sciences, P.O. Box 71007, Wuhan 430071, P.R. China
  • Ezzat S. Noussair

    School of Mathematics, University of New South Wales, Sydney 2052 NSW, Australia
  • Shusen Yan

    Department of Applied Mathematics, South China University of Technology, Guangzhou 510641, China

Abstract

A one to one correspondence is established between the nondegenerate critical points of Q(x) in Ω and single peaked solutions of the problem

where Ω is a bounded domain, 2 < p < (N + 2)/(N - 2), ɛ > 0, and \( Q\left(x\right) \in C\left(\Omega \limits^{¯}\right) \cap C^{2}\left(\Omega \right) \).

In particular, we establish the uniqueness of the least energy solution when Q(x) attains its maximum in \( \Omega \limits^{¯} \) at only one nondegenerate critical point in Ω.

Résumé

On établit une correspondance biunivoque entre les points critiques non-dégénérés de Q(x) en ω, et les solutions à un seul pic du problème

où Ω est un domaine borné, 2 < p < (N + 2)/(N - 2), ɛ > 0, et \( Q\left(x\right) \in C\left(\Omega \limits^{¯}\right) \cap C^{2}\left(\Omega \right) \).

En particulier, nous démontrons l’unicité de la solution de moindre énergie lorsque Q(x) achève son maximum dans Ω en un seul point critique non-dégénéré.

Cite this article

Daomin Cao, Ezzat S. Noussair, Shusen Yan, Existence and uniqueness results on Single-Peaked solutions of a semilinear problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 1, pp. 73–111

DOI 10.1016/S0294-1449(99)80021-3