Solutions Of Semilinear Elliptic Equations Research Articles

Multiple Nontrivial Solutions of Semilinear Elliptic Equations and Their Homotopy Indexes

In this paper, we consider the semilinear elliptic equation −Δu(x) = g(u(x)), x ∈ Ω u(x) = 0, x ∈ ∂Ω, where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω and g : R → R is of class C 2. It is clear that solutions of this problem are equilibria of the local semiflow π g generated by solutions of the semilinear parabolic equation u(t)(t, x) = Δu(t, x) + g(u(t, x)), t ≥ 0, x ∈ Ω u(t, x) = 0, t ≥ 0 x ∈ ∂Ω. We study the existence of multiple nontrivial solutions of the above elliptic equation and their homotopy indices. The homotopy index theory is an extension of Conley′s index theory to noncompact spaces due to Rybakowski and gives some information about the local semiflow π g .

Positive radial solutions of semilinear elliptic equations of order 2m in annular domains

We study the existence of positive radial solutions of (-1)^{m} \Delta^{m}u=g(|x|)f(u) in an annulus with Dirichlet boundary conditions. In par- ticular L^{\infty} a priori bounds are obtained.

Symmetry-breaking at non-positive solutions of semilinear elliptic equations

We consider symmetry-breaking bifurcations at non-positive, radially symmetric solutions of semilinear elliptic equations on a ball with Dirichlet boundary conditions. For nonlinearities which are asymptotically affine linear, we find solutions at which the symmetry breaks. The kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group O(n). For this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough. Thus we obtain, by the equivariant branching lemma, a large variety of isotropy subgroups of O(n) which occur as symmetries of the bifurcating solution branches.

Non-convergent radial solutions of semilinear elliptic equations

Maier, S., Non-convergent radial solutions of semilinear elliptic equations, Asymptotic Analysis 8 (1994) 363-377. 363 We study solutions = u(t) of an initial value problem for u + (n l)/t u' + feu) = 0, which have the additional property that the limit of as t approaches infinity does not exist. Besides some examples, we give necessary conditions on the non-linearity f for the existence of such (non-convergent) solutions. One corollary of these investigations will be that, if f(u)u >0 for small lui '# 0 then every solution u(.;p) (with Ipl small enough) of the initial value problem (1.1), stated below, converges to zero as t -+ 00.

Nonradial solutions of semilinear elliptic equations on annuli

Nonradial solutions of semilinear elliptic equations on annuli

On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions

On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions

Radial Solutions of Semilinear Elliptic Equations with Prescribed Numbers of Zeros

We study solutions u = u( t) of an initial value problem for u′′ + (( n − 1)/ t) u′ + ƒ( u) = 0. Under certain conditions on the nonlinearity ƒ (for instance ƒ( u) = −| u| q + | u| q̂−1 u, 1 < q < q̂ < ( n + 2)/( n − 2), n > 2), we get the existence of some initial values, such that the related solutions converge to zero after having a finite number of zeros. One principal tool to prove this result is derived from the consideration of the relation between critical points and succeeding zeros of solutions of this initial value problem for ƒ( u) = | u| q−1 u + o(| u| q) as | u| → 0.

On the Positive Solutions for Semilinear Elliptic Equations on Annular Domain with Non-homogeneous Dirichlet Boundary Condition

On the Positive Solutions for Semilinear Elliptic Equations on Annular Domain with Non-homogeneous Dirichlet Boundary Condition

Existence of positive solutions for semilinear elliptic equations in annular domains

Existence of positive solutions for semilinear elliptic equations in annular domains

Numerical approximation of bounded solutions for semilinear elliptic equations in an unbounded cylindrical domain

AbstractWe consider the approximation of bounded solutions for a class of semilinear elliptic equations in an unbounded cylindrical domain. On certain conditions, the existence of approximate solutions for the semidiscretization of the problem is proved. Error estimates, applications, and a numerical example are also given. © 1993 John Wiley &amp; Sons, Inc.

Positive solutions of super-critical elliptic equations and asymptotics

This paper is devoted to the study of positive solutions of semilinear elliptic equations . Asymp- totic behavior of ground states and uniqueness of singular ground states are proved via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has one positive solution with fast decay and infinitely many positive solutions with slow decay. The asymptotics of the singular sequence of fast decay solutions when p approaches to is also discussed.

Existence of positive solutions of semilinear elliptic equations in $\mathbb{R}^N$

Existence of positive solutions of semilinear elliptic equations in $\mathbb{R}^N$

‘Large’ solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour

‘Large’ solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour

Existence of multiple solutions to the equations of Monge-Ampère type

In this paper we prove the existence of nontrivial convex solutions to the Monge-Ampère equation det(D 2u) = ƒ(x, u, Du) with Dirichlet or Neumann boundary condition, where ƒ(x, 0, 0) = 0. We also deal with the existence and multiplicity of convex solutions to the equation det(D 2u) = λƒ(x, u, Du) . Our results show that the existence of convex solutions of Mongé-Ampère equations is analogous to that of positive solutions of semilinear elliptic equations.

Nontrivial Solutions of Semilinear Elliptic Equations with Continuous or Discontinuous Nonlinearities

Nontrivial Solutions of Semilinear Elliptic Equations with Continuous or Discontinuous Nonlinearities

On the existence ofG-symmetric entire solutions for semilinear elliptic equations

We prove the existence of at least two solutions of problem (1). The proof is based on the mountain pass theorem and Ekeland’s variational principle.

Uniqueness of Radial Solutions of Semilinear Elliptic Equations

Uniqueness of Radial Solutions 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.

Asymptotic results for finite energy solutions of semilinear elliptic equations

Asymptotic results for finite energy solutions of semilinear elliptic equations

A MINIMIZATION PROBLEM ON EXTERIOR DOMAIN

In this paper, using the decay estimates for the solution of semilinear elliptic equation, we obtain the existence of a minimun for a minimization problem with nonzero data in the exterior domain.