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 of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem:

where

     

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik-Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.     

Multiple positive solutions for a semilinear equation with prescribed singularity

Variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points. Some exact local behavior for positive solutions are also given.     

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.     

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent

In this paper, we study the semilinear p-Laplacian system with critical growth terms in bounded domains. By using the Nehari manifold and variational methods, we prove that the system has at least two positive solutions when the pair of the parameters (λ,μ) belongs to a certain subset of R².     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .

     

Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains

We consider the problem in Ω_ε, u=0 on ∂Ω_ε, where Ω_ε:=ΩB(0,ε) and Ω is a bounded smooth domain in , which contains the origin and is symmetric with respect to the origin, N3 and ε is a positive parameter. As ε goes to zero, we construct sign changing solutions with multiple blow up at the origin.     

Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Lusternik-Schnirelman category to prove that a semilinear elliptic equation involving a sign-changing weight function has at least three positive solutions.     

Multiple solutions for inhomogeneous critical semilinear elliptic problems

In this paper, we consider the existence of multiple positive solutions for an inhomogeneous critical semilinear elliptic problem. We show that the problem possesses at least four positive solutions.     

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^*].     

On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of the nontrivial solutions of the following problem:

     

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.     

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.     

On positive G-symmetric solutions of a weighted quasilinear elliptic equation with critical hardy-sobolev exponent

In this paper, we are concerned with a weighted quasilinear elliptic equation involving critical Hardy–Sobolev exponent in a bounded G-symmetric domain. By using the symmetric criticality principle of Palais and variational method, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on the potential and the nonlinearity.     

Existence results for degenerate elliptic equations with critical cone Sobolev exponents

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.