A global compactness result for singular elliptic problems involving critical Sobolev exponent

Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.

     

Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth

Let be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem in on ∂B_R when ɛ→⁺ for suitable positive numbers μ Mathematics Subject Classification (2000) 35J60, 35B33     

Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

In this paper, we study a class of semilinear elliptic equations with Hardy potential and critical Sobolev exponent. By means of the Ekeland variational principle and Mountain Pass theorem, multiple positive solutions are obtained.     

Three solutions for a nonlocal problem with critical growth

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation ${(?{\mathrm{\Delta }}_p)}^{s}u=|u{|}^{p_s^{?}?2}u+\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where ${(?{\mathrm{\Delta }}_p)}^s$ is the well known p-fractional Laplacian and $p_s^{?}=\frac{np}{n?sp}$ is the critical Sobolev exponent for the non local case. The proof follows the ideas of [28] and is based in the extension of the Concentration Compactness Principle for the p-fractional Laplacian [20] and Ekeland's variational Principle [7].     

A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms

Let be a bounded domain such that . We obtain existence of sign-changing solutions for the Dirichlet problem on Ω,u=0 on ∂Ω for suitable positive numbers μ and λ.     

On a class of singular biharmonic problems involving critical exponents

This paper deals with the following class of singular biharmonic problems

     

Existence result for a class of singular quasilinear elliptic equations with critical growth

In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations with critical exponents. The proofs rely on a non-smooth critical point theory, and some techniques used by Brezis and Nirenberg.     

Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Hardy terms and Hardy-Sobolev critical exponents by using the variational methods.     

含非对称临界非线性项的p-Laplace方程的多解问题

本文考虑含非奇对称临界非线性项的p-Laplace方程Dirichlet问题。运用改进的集中列紧原理证明了在某些指数条件下非奇对称的临界非线性项仍能保证无穷多弱解的存在性。     

Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity

In this paper, we study the multiplicity results of positive solutions for a Kirchhoff type problem with critical growth, with the help of the concentration compactness principle, and we prove that problem admits two positive solutions, and one of the solutions is a positive ground state solution.     

Existence of entire positive solutions for a critical system of nonlinear elliptic equations

In this paper, using the fibering method introduced by Pohozaev, we establish an existence of multiple nontrivial positive solutions for a system of nonlinear elliptic equations in R^N with lack of compactness studying the properties of Palais-Smale sequence of the energy functional associated with the system.     

Positive solutions for critical quasilinear elliptic equations with mixed dirichlet-neumann boundary conditions

The existence and multiplicity of positive solutions are studied for a class of quasilinear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.     

On elliptic problems with two critical Hardy–Sobolev exponents at the same pole

In this paper we study an elliptic problem involving two different critical Hardy–Sobolev exponents at the same pole. By variational methods and concentration compactness principle, we obtain the existence of positive solution to the considered problem.     

Existence of solutions for singular critical semilinear elliptic equation

This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved.     

Sign-changing solutions for elliptic problems with critical Sobolev-Hardy exponents

Let be a smooth bounded domain such that 0∈Ω, N?7, 0?s<2, 2∗(s)=2(N−s)/(N−2). We prove the existence of sign-changing solutions for the singular critical problem −Δu−μ(u/|x|2)=(|u|2∗(s)−2/|x|s)u+λu with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ.     

Solutions for quasilinear elliptic problems with critical Sobolev-Hardy exponents

Let N?3, 2<p<N, 0?s<p and . Via the variational methods and analytic technique, we prove the existence of nontrivial solution to the singular quasilinear problem , for N?p² and suitable functions f(u).     

Infinitely many radial solutions to elliptic problems with critical Sobolev and Hardy terms

Let Ω RN be a ball centered at the origin with radius R > 0 and N 7, 2* = 2N/N-2. We obtain the existence of infinitely many radial solutions for the Dirichlet problem -△u = μ |x|2 u |u|2*-2u λu in Ω, u = 0 on аΩ for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.     

Nonlinear Schrödinger equations with symmetric multi-polar potentials

This paper deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. When the poles form a symmetric structure, it is natural we wonder how the symmetry affects such mutual interaction. The present paper means to study this aspect from the point of view of the existence of solutions inheriting the same symmetry properties as the set of singularities. Mathematics Subject Classification (2000) 35J60, 35J20, 35B33     

Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.