Infinitely many solutions for some nonlinear scalar system of two elliptic equations

In this paper, we consider the elliptic system of two equations in H¹(RN)×H¹(RN):

     

Solutions for semilinear elliptic equations with critical exponents and Hardy potential

In this paper, we answer affirmatively an open problem (cf. Theorem 4′ in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494): Let Ω∋0 be an open-bounded domain, Ω⊂RN(N?5) and assume that , then, for all λ>0 there exists a nontrivial solution with critical level in the range for the problem in Ω; u=0 on ∂Ω.     

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.     

Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

In this paper,a system of elliptic equations is investigated,which involves Hardy potential and multiple critical Sobolev exponents.By a global compactness argument of variational method and a fine analysis on the Palais-Smale sequences created from related approximation problems,the existence of infinitely many solutions to the system is established.     

Infinitely many solutions to elliptic systems with critical exponents and Hardy potentials

In this paper, we consider the following elliptic systems with critical Sobolev growth and Hardy potentials: where N ≥ 3, η > 0, λ₁,λ₂ ∈ [0,Λ_N), and is the best Hardy constant. is the critical Sobolev exponent. a₁, a₂, b₁, and b₂ are positive parameters, and α,β > 1 satisfy 2 < α + β < 2*. h(x) ? 0, h(x) ≥ 0, , , and with . By means of the concentration–compactness principle and R. Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of solutions for the critical elliptic system with inverse square potentials

Let Ω ? 0 be an open bounded domain in R ^N (N ≥ 3) and $2^* (s) = \tfrac{{2(N - s)}} {{N - 2}}$ , 0 < s < 2. We consider the following elliptic system of two equations in H ₀ ¹ (Ω) × H ₀ ¹ (Ω): $$- \Delta u - t\frac{u} {{\left| x \right|^2 }} = \frac{{2\alpha }} {{\alpha + \beta }}\frac{{\left| u \right|^{\alpha - 2} u\left| v \right|^\beta }} {{\left| x \right|^s }} + \lambda u, - \Delta v - t\frac{v} {{\left| x \right|^2 }} = \frac{{2\beta }} {{\alpha + \beta }}\frac{{\left| u \right|^\alpha \left| v \right|^{\beta - 2} v}} {{\left| x \right|^s }} + \mu v,$$ where λ, µ > 0 and α, β > 1 satisfy α + β = 2*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.     

Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: where N ≥ 3,λ₁,λ₂ ∈ [0,Λ_N), is the best Hardy constant. is the critical Sobolev exponent. a₁,a₂, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. with . By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a₁,a₂,b and λ₁,λ₂. Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions to critical elliptic equations with multi-singular inverse square potentials

Let Ω be an open-bounded domain in R^N(N?3) with smooth boundary ∂Ω. We are concerned with the multi-singular critical elliptic problem

     

Infinitely many solutions for non-cooperative elliptic systems

In this paper, we consider the existence of infinitely many solutions of noncooperative elliptic systems perturbed from odd cases.     

Infinitely many solutions for symmetric and non-symmetric elliptic systems

We study an elliptic system equivalent to a fourth order elliptic equation. By using variational and perturbative methods, we prove the existence of infinitely many solutions both in the symmetric and in the non-symmetric case.     

Infinitely many homoclinic solutions for some second order Hamiltonian systems

In this paper, we investigate the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems. By using fountain theorem due to Zou, we obtain two new criteria for guaranteeing that second order Hamiltonian systems have infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.     

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.     

High energy solutions and nontrivial solutions for fourth-order elliptic equations

In this paper, we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth-order elliptic equations in RN via variational methods. Three main theorems are obtained.     

Infinitely many positive solutions for an elliptic problem with critical or supercritical growth

We prove that for some supercritical exponents and for some smooth domains D in RN there are infinitely many (distinct) positive solutions to the following Lane–Emden–Fowler equation: This seems to be the first result for such type of equations.     

具有奇异项的拟线性椭圆型方程组无穷多弱解的存在性

本文在一定条件下讨论了一类具有奇异项的,被两个pLaplacian算子控制的拟线性椭圆型方程组Dirichlet问题无穷多弱解的存在性.     

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:

     

Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems where , , and . The novelty of this paper is that, relaxing the conditions on the potential function W(t,x), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.