Solutions For Semilinear Elliptic Systems Research Articles

The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core

The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Global bifurcation of positive solutions for a class of superlinear elliptic systems

We are concerned with the global bifurcation of positive solutions for semilinear elliptic systems of the form { − Δ u = λ f ( u , v ) in Ω , − Δ v = λ g ( u , v ) in Ω , u = v = 0 on ∂ Ω , where λ ∈ R is the bifurcation parameter, Ω ⊂ R N , N ≥ 2 is a bounded domain with smooth boundary ∂ Ω . We establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ -direction. The nonlinearities f , g ∈ C 1 ( R × R , ( 0 , ∞ ) ) are nondecreasing for each variable and have superlinear growth at infinity. The proof of our main result is based upon bifurcation theory. In addition, as an application for our main result, when f and g subject to the upper growth bound, by a technique of taking superior limit for components, then we may show that the branch must bifurcate from infinity at λ = 0 .

Existence of normalized solutions for semilinear elliptic systems with potential

In this paper, we consider the existence of normalized solutions to the following system: −Δu + V1(x)u + λu = μ1u3 + βv2u and −Δv + V2(x)v + λv = μ2v3 + βu2v in R3, under the mass constraint ∫R3u2+v2=ρ2, where ρ is prescribed, μi, β > 0 (i = 1, 2), and λ∈R appears as a Lagrange multiplier. Then, by a min–max argument, we show the existence of fully nontrivial normalized solutions under various conditions on the potential Vi:R3→R(i=1,2).

Positive solutions for semilinear elliptic systems with boundary measure data

Positive solutions for semilinear elliptic systems with boundary measure data

Nonexistence of nonnegative entire solutions of semilinear elliptic systems

ABSTRACT We consider the second-order semilinear elliptic system where α and β are positive constants, p and q are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions p and q are of slow decay.

Sectional symmetry of solutions of elliptic systems in cylindrical domains

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in [ 6 , 9 , 16 , 18 ].

Multiple positive radial solutions for Neumann elliptic systems with gradient dependence

We provide new results on the existence, nonexistence, and multiplicity of nonnegative radial solutions for semilinear elliptic systems with Neumann boundary conditions on an annulus. Our approach is topological and relies on the classical fixed‐point index. We present an example to illustrate our theory.

Semilinear Elliptic Systems with Dependence on the Gradient

We provide results on the existence, non-existence, multiplicity, and localization of positive radial solutions for semilinear elliptic systems with Dirichlet or Robin boundary conditions on an annulus. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.

Existence and Multiplicity of Solutions for Semilinear Elliptic Systems with Periodic Potential

In this paper, we consider the following semilinear elliptic systems: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$ where $$V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)$$ and $$F_{v}(x,u,v)$$ are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $$-\triangle +V$$ . Under appropriate assumptions on $$F_{u}(x, u, v)$$ and $$F_{v}(x, u, v)$$ , we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended.

A priori bounds and existence of positive solutions for semilinear elliptic systems

We provide a-priori L∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions vp and uq for any (p,q) lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich–Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system.

Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential

We consider a class of semilinear elliptic system of the form: $$\begin{aligned} -\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in {\mathbb {R}}^{2}, \end{aligned}$$ (0.1) where \(W:{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}\) is a double well potential with minima \(\mathbf{a}_\pm \in {\mathbb {R}}^2\). We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system \(-\ddot{q}(x)+\nabla W(q(x))=0,\ x\in {\mathbb {R}}\), up to translations, is finite and constituted by not degenerate functions, then Eq. (0.1) has infinitely many solutions \(u\in C^{2}({\mathbb {R}}^{2})^{2}\), parametrized by an energy value, which are periodic in the variable y and satisfy \(\lim _{x\rightarrow \pm \infty }u(x,y)=\mathbf{a}_{\pm }\) for any \(y\in {\mathbb {R}}\).

Multiple positive solutions for semi-linear elliptic systems involving sign-changing weight on manifolds with conical singularities

In this paper, we use the Nehari manifold method and Ljusternik-Schnirelmann category to prove the multiplicity result of positive solutions for the semi-linear elliptic systems with critical cone Sobolev exponent on manifolds with conical singularities.

Existence and asymptotic behavior of ground state solutions of semilinear elliptic system

Abstract In this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system: -Δu = a 1 a_{1} (x) u α u^{\alpha} v r v^{r} , x ∈ \in ℝ n \mathbb{R}^{n} (n ≥ \geq 3), -Δv = a 2 a_{2} (x) v β v^{\beta} u s u^{s} , x ∈ \in ℝ n \mathbb{R}^{n} , u,v ¿ 0 in ℝ n \mathbb{R}^{n} , lim | x | → + ∞ \lim_{|x|\rightarrow+\infty} u(x) = lim | x | → + ∞ \lim_{|x|\rightarrow+\infty} v(x)=0, where α , β < 1 {\alpha,\beta<1} , r , s ∈ ℝ {r,s\in\mathbb{R}} such that ν := ( 1 - α ) ⁢ ( 1 - β ) - r ⁢ s > 0 {\nu:=(1-\alpha)(1-\beta)-rs>0} , and the functions a 1 a_{1} , a 2 a_{2} are nonnegative in 𝒞 loc γ ⁢ ( ℝ n ) {\mathcal{C}^{\gamma}_{\mathrm{loc}}(\mathbb{R}^{n})} (0¡γ¡1) and satisfy some appropriate assumptions related to Karamata regular variation theory.

Positive solutions for semilinear elliptic systems with sign-changing potentials

Abstract In this paper, we study the existence of positive solutions of the Dirichlet problem -Δu = λ p(x)f(u; v) ; -Δv = λ q(x)g(u; v); in D, and u = v = 0 on ∂∞D, where D ⊂ Rn (n ≥ 3) is an C1,1-domain with compact boundary and λ > 0. The potential functions p; q are not necessarily bounded, may change sign and the functions f; g : ℝ2 → ℝ are continuous with f(0; 0) > 0, g(0; 0) > 0. By applying the Leray- Schauder fixed point theorem, we establish the existence of positive solutions for λ sufficiently small.

Classification of radial solutions for semilinear elliptic systems with nonlinear gradient terms

We are concerned with the classification of positive radial solutions for the system Δu=vp, Δv=f(|∇u|), where p>0 and f∈C1[0,∞) is a nondecreasing function such that f(t)>0 for all t>0. We show that in the case where the system is posed in the whole space RN such solutions exist if and only if ∫1∞(∫0sF(t)dt)−p/(2p+1)ds=∞. This is the counterpart of the Keller–Osserman condition for the case of single semilinear equation. Similar optimal conditions are derived in case where the system is posed in a ball of RN. If f(t)=tq, q>1, using dynamical system techniques we are able to describe the behaviour of solutions at infinity (in case where the system is posed in the whole RN) or around the boundary (in case of a ball).

Uniqueness results for semilinear elliptic systems on Rn

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form As an application we consider the nonlinear Schrödinger system for and exponents q which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi-component nonlinear Schrödinger system is given.

A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities

We prove a priori estimates and existence of positive solutions of semilinear elliptic systems with power nonlinearities and homogeneous Dirichlet boundary conditions. In general, our systems are nonvariational and noncooperative, and our estimates are optimal in the class of very weak solutions. We also provide new Liouville-type theorems.

Existence of multiple positive solutions for semilinear elliptic systems involving m critical hardy-sobolev exponents and m sign-changing weight function

In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

Multiple positive solutions for semi-linear elliptic systems with sign-changing weight

In this paper, we study the multiplicity results of positive solutions for a semi-linear elliptic system involving both concave–convex and critical growth terms. With the help of the Nehari manifold and the Lusternik–Schnirelmann category, we investigate how the coefficient h(x) of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of h.