System Of Semilinear Elliptic Equations Research Articles

Inverse Problem for Equations of Complex Heat Transfer with Fresnel Matching Conditions

An inverse problem is considered for a system of semilinear elliptic equations that simulate radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index. The problem consists in finding the right-hand side of the heat equation, which is a linear combination of given functionals from their specified values on the solution. The solvability of the inverse problem is proved without restrictions on smallness. A sufficient condition for the uniqueness of the solution is presented.

Finding multiple solutions to elliptic systems with polynomial nonlinearity

AbstractElliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L2 norm and H1 norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D4 symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.

Inverse problem with finite overdetermination for steady-state equations of radiative heat exchange

An inverse problem for a system of semilinear elliptic equations modeling simultaneously conductive and radiative heat transfer is under consideration. The problem consists in finding the right-hand side of the heat transfer equation, in the form of linear combination of given functionals, on the base of prescribed values of these functionals on the solution. The solvability of the problem is proven without any smallness assumptions. It is shown that the set of solutions is homeomorphic to a finite-dimensional compact set, and conditions of uniqueness of solutions are found.

The solution structure of the O(3) sigma model in a Maxwell-Chern-Simons theory

In this paper, a system of semilinear elliptic equations arising from a relativistic self-dual Maxwell-Chern-Simons O(3) sigma model is considered. We reveal the uniqueness aspect of the topological solutions for the model. The uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case. We locate each solution set denoted by a planar diagram.

Existence and Nonexistence of Ground State Solutions to Singular Elliptic Systems

In this paper, a system of semi-linear elliptic equations is investigated, which involves multiple critical nonlinearities and multiple singular points. By variational methods, the existence and nonexistence of minimizers to the Rayleigh quotient and ground state solutions to the system are established.

Minimal positive solutions for systems of semilinear elliptic equations

Minimal positive solutions for systems of semilinear elliptic equations

Multidimensional entire solutions for an elliptic system modelling phase separation

For the system of semilinear elliptic equations \[ \Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in $\mathbb{R}^N$} \] we devise a new method to construct entire solutions. The method extends the existence results already available in the literature, which are concerned with the 2-dimensional case, also in higher dimensions $N \ge 3$. In particular, we provide an explicit relation between orthogonal symmetry subgroups, optimal partition problems of the sphere, the existence of solutions and their asymptotic growth. This is achieved by means of new asymptotic estimates for competing system and new sharp versions for monotonicity formulae of Alt-Caffarelli-Friedman type.

Existence of positive solutions to semilinear elliptic systems with supercritical growth

ABSTRACTWe establish the existence of positive entire solutions to cooperative systems of semilinear elliptic equations involving nonlinearities with critical and supercritical growth. Consequently, we obtain existence results to several well-known model examples such as systems of the Hénon, Lane–Emden, and stationary Schrödinger types. The main technique for generating our results relies on a topological approach for the shooting method combined with nonexistence results to closely related boundary value problems.

Minimizers to Rayleigh quotients of critical elliptic systems involving different Hardy-type terms

In this paper, a system of semilinear elliptic equations is investigated, which involves homogeneous critical nonlinearities and different Hardy-type terms. By variational methods, the existence of minimizers to the Rayleigh quotient and ground state solutions to the system is verified completely.

On a Singular Class of Hamiltonian Systems in Dimension Two

Let \Omega be a bounded domain in \mathbb{R}^{2} . In this paper, we consider the following systems of semilinear elliptic equations \text{(S)}\left\{ \begin{alignedat}{2} -\Delta u&=\tfrac{g(v)}{|x|^{a}}&\quad &\hbox{ in }\Omega \\ -\Delta v&=\tfrac{f(u)}{|x|^{b}}&& \hbox{ in }\Omega\\ u&=v=0&& \hbox{ on }\partial\Omega , \end{alignedat} \right. where a,b\in[0, 2) and the functions f and g are nonlinearities having an exponential growth on \Omega . This nonlinearity is motivated by suitable Trudinger-Moser inequality with a singular weight. In fact, we prove the existence of a nontrivial solution to (S). For the proof we use a variational argument (a linking theorem).

Monotone iterates for solving systems of semilinear elliptic equations and applications

Consider monotone finite difference iterative algorithms for solving coupled systems of semilinear elliptic equations. A monotone domain decomposition algorithm based on a modification of Schwarz alternating method and on decomposition of a computational domain into nonoverlapping subdomains is constructed. Advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the nonlinear discrete problem, and is potentially parallelisable. The monotone domain decomposition algorithm is applied to a gas-liquid interaction model. Numerical experiments confirm theoretical results.

Superlinear systems of second-order ODE’s

We discuss the existence of positive solutions of the system − u ″ = f ( t , u , v , u ′ , v ′ ) in ( 0 , 1 ) , − v ″ = g ( t , u , v , u ′ , v ′ ) in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = v ( 0 ) = v ( 1 ) = 0 where the nonlinearities f and g satisfy a superlinearity condition at both 0 and ∞ . Our main result is the proof of a priori bounds for the eventual solutions. As an application, we consider the Dirichlet problem in an annulus for systems of semilinear elliptic equations with nonlinearities depending on the gradient as well. As a second application, we consider fourth-order elastic beam equations with dependence also on the derivatives u ′ , u ″ , u ‴ .

On the existence and nonexistence of solutions for an elliptic system in unbounded domains

In this paper, we study the existence and nonexistence of solutions for some systems of semilinear elliptic equations in an unbounded domain $\Omega$. The main objective of this paper is to discuss the relationship between the existence of solutions and the translation invariance of the domain $\Omega$. More precisely, we point out how the translation invariance of our domain plays a crucial role for the existence of our problem.

On the Topological Multivortex Solutions of the Self-Dual Maxwell-Chern-Simons Gauged O(3) Sigma Model

We consider the self-dual Maxwell-Chern-Simons gauged O(3) sigma model in the whole plane $$\mathbb{R}^2$$ . After reducing to a system of semilinear elliptic equations, we construct a topological solution pair via iteration method and compute the conserved energy explicitly using the exponential decay property of the solution.

On a resonant-superlinear elliptic problem

We start by discussing the solvability of the following superlinear problem \(\) where \(1 < p < \frac{N+1}{N-1}\), \(\Omega \subset \mathbb{R}^N\) is a smooth bounded domain and f satisfies a one-sided Landesman-Lazer condition. We also consider systems of semilinear elliptic equations with nonlinearities of the above form, so exhibiting superlinearity as \(u\to +\infty\) and resonance as \(u\to -\infty\). A priori bounds for the solutions of the equation and the system are obtained by using Hardy-type inequalities . Existence of solutions is then obtained using topological degree arguments.

On the Existence of Many Solutions for a Class of Superlinear Elliptic Systems

We seek multiple solutions to systems of semilinear elliptic equations of the type−ΔU(x)=∇H(x; U(x))x∈ΩU(x)=0x∈∂Ω,where Ω⊂RN (N⩾1) is a bounded regular domain and U=(u1, u2):Ω→R2. For a class of potentials where the variables are strongly coupled as in the model H(x; u1, u2)=|u1|α1|u2|α2 with αi>1, we provide the existence of nine nontrivial solutions characterized by sign properties of each component.

Existence and multiplicity of solutions for systems of semilinear elliptic equations

Existence and multiplicity of solutions for systems of semilinear elliptic equations

General Solutions for Semilinear Elliptic Equations with Electric-Thermal Coupling

This paper deals with a system of semilinear elliptic equations that can represent stationary electric and thermal fields in dissipative media. Due mainly to the exponential form of its conductivity term, the construction of general solutions to the system can be successfully undertaken in cases of practical significance. In all such cases, the general solution can be expressed in terms of Liouville and harmonic scalar fields.

Sublinear elliptic systems with discontinuous nonlinearities

We consider a system of semilinear elliptic equations with a discontinuous nonlinearity which is equivalent to an integro-differential equation.By minimizing the associated energy functional we show the existence of solutions. The regularity of the solutions is also studied.

Positive Solutions of Systems of Semilinear Elliptic Equations: The Pendulum Method

Conditions are formulated which guarantee the existence of positive solutions for systems of the form \[ \begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots , {u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots , {u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots , {u_n}) = {\mu _n}, \hfill \\ \end {gathered} \], where $\Delta$ is the Laplacian (with Dirichlet boundary conditions) on an open domain in ${\mathbf {R}^d}$, and where each ${\mu _i}$ is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where $\partial {f_k}/\partial {x_j} > 0$ for every $k$ and $j$.