Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains

In this paper, we consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, K(x) is a continuous function. When Ω and K(x) are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for 0<λ<λ₁, where λ₁ is the first Dirichlet eigenvalue of on Ω.     

Existence and multiplicity of positive solutions of semilinear elliptic equations in unbounded domains

We investigate the existence and the multiplicity of positive solutions for the semilinear elliptic equation −Δu+u=Q(x)|u|^p−2u in exterior domain which is very close to RN. The potential Q(x) tends to positive constant at infinity and may change sign.     

Existence and multiplicity of positive solutions to elliptic systems involving critical exponents

In this paper, a singular elliptic system involving multiple critical exponents and the Caffarelli-Kohn-Nirenberg inequality is investigated. By using the extremals of the best Hardy-Sobolev constants, the existence and multiplicity of positive solutions to the system are established.     

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.     

Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents

In this paper, we study the existence and multiplicity of positive solutions to the following system , in Ω; u,v>0 in Ω; and u=v=0 on ∂Ω, where Ω is a bounded smooth domain in R^N; FC¹((R⁺)²,R⁺) is positively homogeneous of degree μ; ; and ε is a positive parameter. Using sub–supersolution method, we prove the existence of positive solutions for the above problem. By means of the variational approach, we prove the multiplicity of positive solutions for the above problem with μ(2,2^*].     

Nonexistence of multi-bubble solutions to some elliptic equations on convex domains

We prove the nonexistence of multi-bubble solutions for several types of problems on smooth bounded convex domains. Problems we study include the Liouville equation

     

Elliptic systems involving multiple critical nonlinearities and symmetric multi-polar potentials

In this paper,a system of elliptic equations is investigated,which involves multiple critical Sobolev exponents and symmetric multi-polar potentials.By variational methods and analytic techniques,the relevant best constants are studied and the existence of(Zk×SO(N.2))2-invariant solutions to the system is established.     

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.     

Existence and multiplicity of positive solutions for some Hardy-Sobolev critical elliptic equation with boundary singularities

In this paper, by Ekeland’s variational principle and strong maximum principle, we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical Hardy-Sobolev exponents and Hardy terms with boundary singularities.