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Mountain pass solutions for perturbed Hardy–Sobolev equations on compact manifolds

  • Hassan Jaber EMAIL logo
Published/Copyright: March 19, 2016
Analysis
From the journal Analysis Volume 36 Issue 4

Abstract

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥3. We fix a point x0∈M and s∈(0,2). We investigate a sufficient condition for the existence of a distributional continuous positive solution for the Hardy–Sobolev equation

Δg⁢u+a⁢(x)⁢u=u2⋆⁢(s)-1dg⁢(x,x0)s+h⁢uq-1 in ⁢M,

where Δg:=-divg⁡(∇) is the Laplace–Beltrami operator, a,h∈C0⁢(M), h≥0, dg is the Riemannian distance on (M,g), 2⋆⁢(s)=2⁢(n-s)n-2, and q∈(2,2⋆) with 2⋆=2⋆⁢(0). We prove that the existence of a mountain pass solution for the above perturbative equation depends only on the perturbation when n≥4, while for n=3, it depends on other conditions involving the perturbation and the global geometry of the manifold.

Keywords: Compact Riemannian manifold; mountain pass solution; Hardy–Sobolev equation; mass of manifold; mountain pass lemma
MSC 2010: 35-01; 35A01; 35B50; 35J15; 35D35; 53-01; 53C21; 58J05

Funding statement: The author was partially supported by Institut Eli Cartna de Lorraine, IECL (UMR 7502) and Centre de Physique Théorique de Marseille, CPT (UMR 7332).

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Received: 2016-1-20
Accepted: 2016-3-1
Published Online: 2016-3-19
Published in Print: 2016-11-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Investigations about the Euler–Mascheroni constant and harmonic numbers
  3. A class of subsums of Euler’s sum
  4. Lyapunov type inequalities for second-order differential equations with mixed nonlinearities
  5. Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications
  6. Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions
  7. The time-dependent Stokes problem with Navier slip boundary conditions on Lp-spaces
  8. Mountain pass solutions for perturbed Hardy–Sobolev equations on compact manifolds
https://doi.org/10.1515/anly-2016-0004
Keywords for this article
Compact Riemannian manifold; mountain pass solution; Hardy–Sobolev equation; mass of manifold; mountain pass lemma

Articles in the same Issue

  1. Frontmatter
  2. Investigations about the Euler–Mascheroni constant and harmonic numbers
  3. A class of subsums of Euler’s sum
  4. Lyapunov type inequalities for second-order differential equations with mixed nonlinearities
  5. Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications
  6. Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions
  7. The time-dependent Stokes problem with Navier slip boundary conditions on Lp-spaces
  8. Mountain pass solutions for perturbed Hardy–Sobolev equations on compact manifolds
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