On domains with conical points, weighted Sobolev spaces with powers of the distance to the conical points as weights form a classical framework for describing the regularity of solutions of elliptic boundary value problems, cf. papers by Kondrat’ev and Maz’ya-Plamenevskii. Two classes of weighted norms are usually considered: Homogeneous norms, where the weight exponent varies with the order of...

We consider the problem: −div(p∇u) = u + λu, u > 0 in Ω, u = 0 on ∂Ω. Where Ω is a bounded domain in IR, n ≥ 3, p : Ω̄ −→ IR is a given positive weight such that p ∈ H(Ω) ∩ C(Ω̄), λ is a real constant and q = 2n n−2 . We study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.

Let (M, g) be a smooth, compact Riemannian n-manifold, and h be a Hölder continuous function on M . We prove the existence of multiple changing sign solutions for equations like ∆gu + hu = |u| ∗−2 u, where ∆g is the Laplace–Beltrami operator and the exponent 2∗ = 2n/ (n− 2) is critical from the Sobolev viewpoint.

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions for the problem is obtained. The result depends crucially on the parameters p, t, s, λ and μ. c © 2007 Elsevier Ltd. All rights reserved. MSC: 35J60; 35B33

For the equation −∆u = ||x| − 2| α u p−1 , 1 < |x| < 3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2 * = 2N/(N − 2). A symmetry– breaking phenomenon appears, showing that the least–energy solutions cannot be radial functions.

In this article, we show the existence of multiple positive solutions to a class of degenerate elliptic equations involving critical cone Sobolev exponent and sign-changing weight function on singular manifolds with the help of category theory and the Nehari manifold method.

In this article, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problems involving critical Sobolev exponent and a Hardy term. The main tools adopted in our proofs are the concentration compactness principle and Nehari manifold.

We prove that the best constant in the Sobolev inequality (WI,” c Lp* with $= f i and 1 c p < n) is achieved on compact Riemannian manifolds, or only complete under some hypotheses. We also establish stronger inequalities where the norms are to some exponent which seems optimal. 0 Elsevier, Paris

We study the non-linear minimization problem on H 0 (Ω) ⊂ L q with q = 2n n−2 : inf ‖u‖ Lq =1 ∫ Ω (1 + |x| |u|)|∇u|. We show that minimizers exist only in the range β < kn/q which corresponds to a dominant nonlinear term. On the contrary, the linear influence for β ≥ kn/q prevents their existence.