JournalsaihpcVol. 13, No. 2pp. 185–227

Some sufficient conditions for the existence of positive solutions to the equation −∆u + a(x)u = u2*−1 in bounded domains

  • Donato Passaseo

    Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti, 2, 56127 – PISA
Some sufficient conditions for the existence of positive solutions to the equation −∆u + a(x)u = u2*−1 in bounded domains cover
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Abstract

This paper is concerned with the problem

{Δu+a(x)u=un+2n2inΩu>0inΩ;u=0onΩ\left\{\begin{matrix} −\mathrm{\Delta }u + a(x)u = u^{\frac{n + 2}{n−2}} & \mathrm{in}\:Ω \\ u > 0\:\:\mathrm{in}\:\:Ω;\:\:\:u = 0 & \mathrm{on}\:∂Ω \\ \end{matrix}\right.

where Ω is a bounded domain in ℝn with n ≥ 3 and a(x) is a nonnegative function in Ω. We give some conditions on the function a(x), sufficient to guarantee the existence and multiplicity of solutions for the considered problem without any assumption on the shape of Ω.

Résumé

On considère le problème (*) où Ω est un ouvert borné de ℝn avec n ≥ 3 et a(x) une fonction non-négative dans Ω.

On établit des conditions sur la fonction a(x) suffisantes pour assurer l’existence et la multiplicité de solutions du problème considéré sans aucune condition sur la forme de Ω.

Cite this article

Donato Passaseo, Some sufficient conditions for the existence of positive solutions to the equation −∆u + a(x)u = u2*−1 in bounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 2, pp. 185–227

DOI 10.1016/S0294-1449(16)30102-0