Solutions For Semilinear Elliptic Systems Research Articles

Multiple solutions of semilinear elliptic systems on the Heisenberg group

In this paper, a class of semilinear elliptic systems which have a strong resonance at the first eigenvalue on the Heisenberg group is considered. Under certain assumptions, by virtue of the variational methods, the multiple weak solutions of the systems are obtained.MSC:35J20, 35J25, 65J67.

A remark on the radial and nonradial solutions for an elliptic system with a quadratic gradient term

Recently, the authors Yang and Zhang in [Y. Yang, X. Zhang, Entire blow-up solutions of semilinear elliptic system with quadratic gradient term, Abstract and Applied Analysis, 2012] have been obtained sufficiently conditions for the existence and non-existence of radially symmetric solutions for the system of equations Δui+∣∇ui∣2=ai(x)fi(u1,…,ud) on RN (N⩾3) with i=1,…,d and d∈{1,2,3,4,…}. Our aim in this paper is to give a necessary and a sufficient condition, but for the existence of large (or explosive) solutions for the same class of systems. Moreover, we intend to obtain a non-radial solution for the considered problem. We also point out that our framework in the obtained results improve and complete some of the results obtained by Yang and Zhang.

Multiplicity of Positive Solutions for Semilinear Elliptic Systems

We study the effect of the coefficient of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for , , and , we should prove that for sufficiently small , there are at least positive solutions of the semilinear elliptic systems , , where is a bounded domain, , , and for .

Existence and multiplicity of solutions for semilinear elliptic systems involving Hardy-Sobolev critical nonlinearity

This paper is concerned with a singular elliptic system, which involves the Hardy- Sobolev critical nonlinearity. The existence and multiplicity of solutions for this system are obtained by the variational methods.

Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in R N

In this paper, we investigate the effect of the coefficient f(x) of the subcritical nonlinearity. Under some assumptions, for sufficiently small ε,λ,μ>0, there are at least k (≥1) positive solutions of the semilinear elliptic systems { − ε 2 Δ u ¯ + u ¯ = λ g ( x ) | u ¯ | q − 2 u ¯ + α α + β f ( x ) | u ¯ | α − 2 u ¯ | v ¯ | β in R N ; − ε 2 Δ v ¯ + v ¯ = μ h ( x ) | v ¯ | q − 2 v ¯ + β α + β f ( x ) | u ¯ | α | v ¯ | β − 2 v ¯ in R N ; u ¯ , v ¯ ∈ H 1 ( R N ) , where α>1, β>1, 2<q<p=α+β< 2 ∗ =2N/(N−2) for N≥3.MSC:35J20, 35J25, 35J65.

Existence of entire positive solutions for semilinear elliptic systems with gradient term

Under the Keller–Osserman condition on \({\Sigma_{j=1}^{2}f_{j}}\), we show the existence of entire positive solutions for the semilinear elliptic system \({\Delta u_{1}+|\nabla u_{1}|=p_{1}(x)f_{1}(u_{1},u_{2}), \Delta u_{2}+|\nabla u_{2}|=p_{2}(x)f_{2}(u_{1},u_{2}),x \in \mathbb{R}^{N}}\), where \({p_{j}(j=1, 2):\mathbb{R}^{N} \rightarrow [0,\infty)}\) are continuous functions.

Existence of semiclassical ground-state solutions for semilinear elliptic systems

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε &gt; 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

Existence of non-negative solutions for semilinear elliptic systems via variational methods

In this paper we consider a semilinear elliptic system with nonlinearities, indefinite weight functions and critical growth terms in bounded domains. The existence result of nontrivial nonnegative solutions is obtained by variational methods.

Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient f(z) of the critical nonlinearity. For sufficiently small λ,μ>0, there are at least k positive solutions of the semilinear elliptic systems{−Δu=λg(z)|u|p−2u+αα+βf(z)|u|α−2u|v|βin Ω;−Δv=μh(z)|v|p−2v+βα+βf(z)|u|α|v|β−2vin Ω;u=v=0on ∂Ω, where 0∈Ω⊂RN is a bounded domain, α>1, β>1 and 2<p<α+β=2⁎ for N>4.

Symmetry results for decay solutions of semilinear elliptic systems on half spaces

In this paper, we prove that the decaying positive solutions of semi-linear elliptic systems on the half space with Neumann type condition is symmetric with respect to a line orthogonal to the boundary hyperplane. The monotonicity result is also obtained. Our methods employ the Alexandrov–Serrin method of moving planes combined with Hopf’s lemma at a corner. Our result can be applied to the Lane–Emden system and the stationary Schrödinger system.

Entire Blow‐Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, Δui+|∇ui|2 = pi(|x|)fi(u1, u2, …, ud) for i = 1,2, …, d on RN, N ≥ 3 and d ∈ {1,2, 3, …}. We establish the conditions on pi that ensure the existence of nonnegative radial solutions blowing up at infinity and also the conditions for bounded solutions on the entire space. The condition on fi is simple and different to the Keller‐Osserman condition.

The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term

We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δ u + | ∇ u | = p ( | x | ) f ( u , v ) , Δ v + | ∇ v | = q ( | x | ) g ( u , v ) on R N , N ⩾ 3 , provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

Solutions for Semilinear Elliptic Systems with Critical Sobolev Exponent and Hardy Potential

AbstractIn this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of ℝN. By variational methods we study the existence results.

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 − Δ u = ∂ F ∂ u ( u , v ) + ε g ( x ) , − Δ v = ∂ F ∂ v ( u , v ) + ε h ( x ) in Ω ; u , v > 0 in Ω ; and u = v = 0 on ∂ Ω , where Ω is a bounded smooth domain in R N ; F ∈ C 1 ( ( R + ) 2 , R + ) is positively homogeneous of degree μ ; g , h ∈ C 1 ( Ω ¯ ) ∖ { 0 } ; 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 ∗ ] .

Existence and multiplicity of solutions for a class of elliptic systems in [formula omitted

Using minimax methods we study the existence and multiplicity of nontrivial solutions for semilinear elliptic systems of the form − Δ u i + a i ( x ) u i = f i ( x , u 1 , … , u m ) for x ∈ R n and i = 1 , … , m . The potentials a i ( x ) for i = 1 , … , m may change sign, control the growth of nonlinearities and can be “large” at the infinity. We treat both superquadratic situation and nonquadratic situation at infinity on the nonlinearities f i ( x , u 1 , … , u m ) for i = 1 , … , m .

Nonexistence and Radial Symmetry of Positive Solutions of Semilinear Elliptic Systems

Nonexistence and radial symmetry of positive solutions for a class of semilinear elliptic systems are considered via the method of moving spheres.

Existence of positive entire solutions for semilinear elliptic systems in the whole space

This work is devoted to the existence ofpositive entire solutions for semilinear elliptic systems. With theaid of a degree theory argument, we use the shooting method andPohozaev-type identity to show the existence of positive radialsolutions.

Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems

Existence, localization and multiplicity results of positive solutions to a system of singular second-order differential equations are established by means of the vector version of Krasnoselskii's cone fixed point theorem. The results are then applied for positive radial solutions to semilinear elliptic systems.

Existence results and a priori bounds for higher order elliptic equations and systems

We apply degree theory to prove the existence of positive solutions of semilinear elliptic systems. As an application we obtain a number of new results for higher order equations which appear frequently in applications. In particular, we extend to these equations and systems the notions of sublinearity and superlinearity, classical in the setting of second order equations.

Multiplicity of solutions for a class of non-homogeneous fourth-order boundary value problems

We study the multiplicity of positive solutions for a class of fourth-order boundary value problems with non-homogeneous boundary conditions. For this, we use a fixed point theorem of cone expansion/compression type and we establish a general theorem for a type of systems of second-order ordinary differential equations involving parameters. In addition, we apply our result to the study of the existence of solutions for semilinear elliptic systems in bounded annular domains.