Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

  • Vladimir Kondratiev

    Department of Mathematics and Mechanics, Moscow State University, Moscow 119 899, Russia
  • Vitali Liskevich

    School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
  • Vitaly Moroz

    School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Abstract

We study the problem of the existence and nonexistence of positive solutions to the superlinear second-order divergence type elliptic equation with measurable coefficients −\mathrm{∇} \cdot a \cdot \mathrm{∇}u = u^{p}$$(*), p>1p > 1, in an unbounded cone-like domain G \subset \mathbb{R}^{N}$$(N⩾3). We prove that the critical exponent p(a,G)=inf{p>1:()has a positive supersolution at infinity inG}p^{*}(a\text{,}G) = \mathrm{\inf }\{p > 1:\:(*)\:\text{has a positive supersolution at infinity in}\:G\} for a nontrivial cone-like domain is always in (1,NN2)(1\text{,}\frac{N}{N−2}) and depends both on the geometry of the domain G and the coefficients a of the equation.

Résumé

Nous étudions le problème d'existence ou non existence de solutions positives d'équations elliptiques de type divergence du second-ordre superlinéaires à coefficients mesurables −\mathrm{∇} \cdot a \cdot \mathrm{∇}u = u^{p}$$(*), p>1p > 1, sur un domaine conique G \subset \mathbb{R}^{N}$$(N⩾3). Nous prouvons que l'exposant critique p(a,G)=inf{p>1:()a une supersolution positive aˋ l’infini dansG}p^{*}(a\text{,}G) = \mathrm{\inf }\{p > 1:\:(*)\:\text{a une supersolution positive à l'infini dans}\:G\} pour un domaine conique non–trivial est toujours dans (1,NN2)(1\text{,}\frac{N}{N−2}), dépend à la fois de la géometrie du domaine G et des coefficients a de l'équation.

Cite this article

Vladimir Kondratiev, Vitali Liskevich, Vitaly Moroz, Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 1, pp. 25–43

DOI 10.1016/J.ANIHPC.2004.03.003