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.