6533b872fe1ef96bd12d302d
RESEARCH PRODUCT
Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents
Donato FortunatoMichael StruweGiovanna Ceramisubject
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 eigenvectorsBifurcationMathematicsdescription
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 manifold of dimension n ⩾ 3, without boundary and Δ is the relative Laplace-Beltrami operator.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 1984-10-01 |