Semilinear Elliptic Systems Research Articles

Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain

Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain

Existence of solutions for some semilinear elliptic systems with singular coefficients

Existence of solutions for some semilinear elliptic systems with singular coefficients

On a Class of Semilinear Elliptic Systems and Applications in Polyharmonic Equations

In this paper we study a class of semilinear differential equation systems. The boundedness of positive solutions of the systems has been shown under some general assumptions. We give some applications for the systems; in particular, with these results, we prove that any positive solution of some polyharmonic equations involving critical exponents is also a poly-super-harmonic function.

On critical semilinear elliptic systems

We establish in this paper existence results for critical strongly indefinite semilinear elliptic systems defined on both bounded domains and $\mathbb R^N$.

The absence of global positive solutions of systems of semilinear elliptic inequalities in cones

Let be a cone in , . We establish conditions for the absence of global non-trivial non-negative solutions of semilinear elliptic inequalities and systems of inequalities of the form We find the critical exponent that divides the domains of existence of these solutions from those of their absence. We prove that in the limiting case there are no solutions. The method is to multiply the system by a special factor and integrate the inequalities thus obtained.

Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems

We show that entire large positive radial solutions exist for the semilinear elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on RN, N⩾3, for positive α and β, provided that the nonnegative functions p and q are continuous, c-positive, and satisfy the decay conditions ∫∞0tp(t)dt<∞ and ∫∞0tq(t)dt<∞ for α and β greater than unity, and ∫∞0tp(t)dt=∞ and ∫∞0tq(t)dt=∞ if neither α nor β is greater than one.

Symmetry Results for Semilinear Elliptic Systems in the Whole Space

In this paper we obtain symmetry results of Gidas-Ni-Nirenberg type for cooperative nonlinear elliptic systems in the whole space

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

It is shown that there exists Λ>0 such that, for every λe(0, Λ), the semilinear elliptic system: Δu = λu | u |q-1 + u | u |p-1 - υ, in ω, u=v=0 on ∂ ω, where ω eR N(N≥2) is a bounded domain with smooth boundary and 0<q<1<p, has a minimal positive solution (uλ, vλ). Moreover: uλ and vλ are strictly increasing with respect to λ.

Non-Existence Results for Semilinear Cooperative Elliptic Systems via Moving Spheres

Based upon a new development of the method of moving spheres, we introduce a new and general approach for non-existence of positive solutions of cooperative semilinear elliptic systems with the Laplacian as principal part. For supercritical nonlinearities we prove non-existence on bounded star-shaped domains. For subcritical nonlinearities we obtain non-existence results on a class of unbounded domains, which includes e.g. the entire space, certain curved halfspaces and the complement of bounded star-shaped domains. As a by-product we also get a symmetry result on halfspaces.

Local behaviour of the solutions of a class of nonlinear elliptic systems

Here we study the behaviour near a punctual singularity of the positive solutions of semilinear elliptic systems in ${\mathbb R}^{N}(N\geq 3)$ given by \[ \left\{ \begin{array}{c} \Delta u+\left| x\right| ^{a}u^{s}v^{p}=0, \\ \Delta v+\left| x\right| ^{b}u^{q}v^{t}=0, \end{array} \right. \] (where $a,b,p,q,s,t\in $ ${\mathbb R}$ , $p,q>0,s,t\geq 0$). We describe the first undercritical case, and the sublinear and linear cases. The proofs do not use any variational methods, but lie essentially upon comparison properties between the two solutions $u$ and $v$, and the properties of the subsolutions and supersolutions of the scalar equation \[ \Delta f+\left| x\right| ^{\sigma }f^{\eta }=0 \] ($\sigma ,\eta \in $ ${\mathbb R}$ , $\eta >0$). This extends the classical study of the scalar equation when $0 <\eta <\max $ $(N,(N+\sigma ))/(N-2)$.

Existence and uniqueness of positive solutions of semilinear elliptic systems

Existence and uniqueness of positive solutions of semilinear elliptic systems

Zero sets of solutions to semilinear elliptic systems of first order

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an $n$-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth $(n-2)$-dimensional submanifolds. Hence it is countably $(n-2)$-rectifiable and its Hausdorff dimension is at most $n-2$. Moreover, it has locally finite $(n-2)$-dimensional Hausdorff measure. We show by example that every real number between 0 and $n-2$ actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.

Symmetry of ground states for a semilinear elliptic system

Let n ≥ 3 n\ge 3 and consider the following system Δ u + f ( u ) = 0 , u &gt; 0 , x ∈ R n . \begin{equation*} \Delta \mathbf {u}+\mathbf {f}(\mathbf {u})=0,\quad \mathbf {u}&gt;0,\qquad x\in \mathbf {R}^n.\end{equation*} By using the Alexandrov-Serrin moving plane method, we show that under suitable assumptions every slow decay solution of (I) must be radially symmetric.

Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system

Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system

A boundary layer solution to a semilinear elliptic system of FitzHugh-Nagumo type

In this Note we study the semilinear elliptic system (P λ) on a bounded domain, where λ, δ, γ > 0 and ƒ as two consecutive zeroes, for example ƒ(u) = u(u Å)(l u) with 0 < a < 1/2. The existence, for λ large, of a positive solution with boundary layer behavior as well as uniqueness of this solution in an order interval are proven.

A priori estimates for positive solutions for a class of semilinear elliptic systems

A priori estimates for positive solutions for a class of semilinear elliptic systems

Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation, namely a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second order optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second order sufficient condition holds, we give a formula for the second order expansion of the value of the problem as well as the directional derivative of the optimal control, when the cost function is perturbed. Then we partially extend these results to the case of vector valued controls when the feasible set is defined by local and smooth convex constraints. When the space dimension $n$ is greater than 3, the results are based on a two norms approach, involving spaces $L^2(Om)$ and $L^s(Om)$, with $s>n/2$.

Monotonicity and convergence results in order-preserving systems in the presence of symmetry

This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.

Decay, symmetry and existence of solutions of semilinear elliptic systems

Our utmost aim was originally to establish the existence of solutions of system. However, in this effort we got sidetracked to consider other questions. In this way we come to interesting results on asymptotic behavior of solutions, on symmetry properties of such solutions, and the existence of ground states.

The numerical solution of some elliptic nonscalar problems with nonlinear discontinuities

This paper is devoted to the numerical solution of some semilinear elliptic systems with nonlinear discontinuities. The motivation is the modelling of single, irreversible, nonisothermic stationary reactions of zero order. The numerical solution is achieved through an iterative procedure. This uses exact regularization. We justify the convergence of the algorithm and, also, we present some numerical results.