Search results for "Sobolev exponent"

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Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents

1984

Abstract In this paper we study the existence of nontrivial solutions for the boundary value problem { − Δ u − λ u − u | u | 2 ⁎ − 2 = 0 in Ω u = 0 on ∂ Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2 ⁎ = 2 n ( n − 2 ) is the critical exponent for the Sobolev embedding H 0 1 ( Ω ) ⊂ L p ( Ω ) , λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λ j ⁎ , λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ ∈ ] λ j ⁎ , λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manif…

Pure mathematicsRiemannian manifoldApplied MathematicsMathematical analysisEigenvalueCritical Sobolev exponentMultiplicity (mathematics)Critical pointsRiemannian manifoldSobolev spaceBounded functionBoundary value problem; Critical Sobolev exponent; Bifurcation; Critical points; Eigenvalue; Variational problem; Riemannian manifoldBifurcationVariational problemBoundary value problemCritical exponentBoundary value problemMathematical PhysicsAnalysisEigenvalues and eigenvectorsBifurcationMathematics
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Isotropic stochastic flow of homeomorphisms on Sd for the critical Sobolev exponent

2006

Abstract In this work, we shall deal with the critical Sobolev isotropic Brownian flows on the sphere S d . Based on previous works by O. Raimond and LeJan and Raimond (see [O. Raimond, Ann. Inst. H. Poincare 35 (1999) 313–354] and [Y. LeJan, O. Raimond, Ann. of Prob. 30 (2002) 826–873], we prove that the associated flows are flows of homeomorphisms.

Sobolev exponentKolmogoroff modification theoremApplied MathematicsGeneral MathematicsEigenvectorIsotropyMathematical analysisSpherical representationHomeomorphismNon-Lipschitzian conditionSobolev spacesymbols.namesakeLaplace operatorMathematics::ProbabilityPoincaré conjecturesymbolsExponentIsotropic flowsLaplace operatorCritical exponentBrownian motionMathematicsJournal de Mathématiques Pures et Appliquées
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