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

Daomin Cao; Ezzat S. Noussair; Shusen Yan

doi:10.1016/s0294-1449(99)80021-3

JournalsaihpcVol. 15, No. 1pp. 73–111

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

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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

- ɛ^{2} Δ u + u = Q (x) u^{p - 1} u > 0 in Ω and u = 0 in Ω on \partial Ω

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

- ɛ^{2} Δ u + u = Q (x) u^{p - 1} u > 0 dans Ω et u = 0 dans Ω sur \partial Ω

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