Semilinear Elliptic Partial Differential Equations Research Articles

The Finite Element Approximation of Semilinear Elliptic Partial Differential Equations with Critical Exponents in the Cube

We consider the finite element solution of the parameterized semilinear elliptic equation $\Delta u + \lambda u + u^{5} = 0, u > 0$, where $u$ is defined in the unit cube and is 0 on the boundary of the cube. This equation is important in analysis, and it is known that there is a value $\lambda_{0} > 0$ such that no solutions exist for $\lambda \lambda_{0}$ and for the form of the spurious numerical solutions which are known to exist when $\lambda < \lambda_{0}$. These estimates are then used to post-process the numerical results to obtain a sharp estimate for $\lambda_{0}$ which agrees with the conjectured value.

Uniqueness of Positive Solutions of Δ u + f ( u )=0 in ℝ N , N ≦3

We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation $$\Delta u+f(u)=0 \eqno{(*)}$$ in ℝ N . We assume that the function f has two zeros, the origin and u 0>0. Above u 0 the function f is positive, is locally Lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature. Below u 0, f is assumed to be non-positive, non-identically zero and merely continuous. Our results are obtained through a careful analysis of the solutions of an associated initial‐value problem, and the use of a monotone separation theorem. It is known that, for a large class of functions f, the ground states of (*) are radially symmetric. In these cases our result implies that (*) possesses at most one ground state.

Internal layers in high-dimensional domains

For a system of semilinear elliptic partial differential equations with a small parameter, denned on a bounded multi-dimensional smooth domain, we show the existence of solutions with internal layers. The high-dimensionality of the domain gives rise to quite interesting an outlook in the analysis, dramatically different from that in one-dimensional settings. Our analysis indicates, in a certain situation, an occurrence of an infinite series of bifurcation phenomena accumulating as the small parameter goes to zero. We also present a related free boundary problem with a possible approach to its resolution.

Mountain Pass on a Closed Convex Set

A mountain pass lemma without the Palais–Smale condition on a closed convex subset of a Banach space is established. Then, we apply it to a semilinear elliptic partial differential equation to obtain one negative solution and a positive solution. Hence we generalize an early result of K. C. Chang and L. Nirenberg.

Boundary element monotone iteration scheme for semilinear elliptic partial differential equations

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin δ \delta for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the H r ( Ω ) H^{r}(\Omega ) , 0 ≤ r ≤ 2 0\le r \le 2 , Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, “higher than optimal order” error estimates can be obtained with respect to the mesh parameter h h . Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.

A SEMI-LINEAR ELLIPTIC PDE MODEL FOR THE STATIC SOLUTION OF JOSEPHSON JUNCTIONS

In this study we derive a semi-linear Elliptic Partial Differential Equation (PDE) problem that models the static (zero voltage) behavior of a Josephson window junction. Iterative methods for solving this problem are proposed and their computer implementation is discussed. The preliminary computational results that are given, show the modeling power of our approach and exhibit its computational efficiency.

Solutions with spikes for quasilinear boundary value problems

In recent years there has been some progress in the study of spike layer solutions for semilinear singular perturbation problems. Spike layer behavior can be described as nonuniform limiting behavior of a solution in which the solution has an interior maximum or minimum inside the layer. One of the first rigorous studies was made by O’Malley [lo], who used phase plane analysis for the autonomous semilinear Dirichlet problem and showed that solutions exist with increasing numbers of spikes as the small parameter approaches zero and that the spikes occur at equally spaced points in the interval. More recently, various results for nonautonomous ordinary differential equations with Dirichlet or Neumann boundary conditions have been obtained by Kath [4], Lange [7], Butuzov and Vasil’eva [ 11, and Kurland [6]. Spike layer solutions for semilinear elliptic partial differential equations have been treated by Lin, Ni, and Takagi [8] for Neumann boundary conditions and by Kelley and Ko [S] for Dirichlet boundary conditions. We consider in this paper the quasilinear problem

An extended equivariant branching theory

AbstractRecent monographs and research articles have demonstrated that the highly complicated bifurcation structure, which usually arises in the presence of high symmetries, can often be systematically reduced with group theoretic concepts. Equivariant branching theorems and bifurcation subgroups have been the principle tools in this direction of investigation. The generalizations that we give allow the discussions of new classes of problems and the weakening of the usual hypotheses of absolute irreducibility of the group representation or of the appropriate subspaces. These extensions are illustrated in the case of a class of planar semilinear elliptic partial differential equations, which are usually treated as model problems in fluid mechanics and chemical reactions.

On the Perturbation of Critical Values of Maximum-Minimum Type

The stability of critical values of a functional under small perturbations is investigated. There are considered critical values of maximum-minimum type of functionals which are related to boundary value and cigenvalue problems for semilinear elliptic partial differential equations.

Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints

Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints

Asymptotics for Semilinear Elliptic Systems

AbstractA class of weakly coupled systems of semilinear elliptic partial differential equations is considered in an exterior domain in ℝN, N &gt; 3. Necessary and sufficient conditions are given for the existence of a positive solution (componentwise) with the asymptotic decay u(x) = O(|x|2-N) as |x| —&gt; ∞. Additional results concern the existence and structure of positive solutions u with finite energy in a neighbourhood of infinity.

Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN

SynopsisWe consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

Monotone iterations for nonlinear obstacle problem

AbstractThe existence, uniqueness and regularity of solutions are proved for the obstacle problem with semilinear elliptic partial differential equations of second order. Computationally effective algorithms are provided and application made to steady state problem for the logistic population model with diffusion and an obstacle to growth.

Nonlinear second order elliptic partial differential equations at resonance

In this paper we study the solvability of boundary value problems for semilinear second order elliptic partial differential equations of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy the Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded and some crossing of eigenvalues is allowed. Selfadjoint and nonselfadjoint resonance problems are considered.

Radial Entire Solutions of Even Order Semilinear Elliptic Equations

Semilinear elliptic partial differential equations of the type1will be considered throughout real Euclidean N-space, where m ≧ 2 is a positive integer, Δ denotes the N-dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of |x|2m−2i as |x| → ∞ for every i = 1,…, m, N ≧ 2i + 1.

On a class of functionals invariant under a 𝑍ⁿ action

Consider a system of ordinary differential equations of the form ( ∗ ) ({\ast }) \[ q ¨ + V q ( t , q ) = f ( t ) \ddot q + {V_q}(t,\,q) = f(t) \] where f f and V V are periodic in t t , V V is periodic in the components of q = ( q 1 , … , q n ) q = ({q_1}, \ldots ,{q_n}) , and the mean value of f f vanishes. By showing that a corresponding functional is invariant under a natural Z n {{\mathbf {Z}}^n} action, a simple variational argument yields at least n + 1 n + 1 distinct periodic solutions of (*). More general versions of (*) are also treated as is a class of Neumann problems for semilinear elliptic partial differential equations.

Solution, Gradient, and Laplacian Bounds in Some Nonlinear Fourth Order Elliptic Equations

In order to obtain bounds of the type in the title, a suitable function is defined on the solutions to a certain class of semilinear fourth order elliptic partial differential equations. The subharmonicity of this function under appropriate conditions on the coefficients and nonlinear terms leads one immediately to the desired bounds.

Accelerated monotone scheme for finite difference equations concerning steady-state prey–predator interactions

In previous work, by adapting a suitable finite difference method to a particular monotone scheme, the authors and A. Lazer have studied the numerical solution of a system of semilinear elliptic partial differential equations which determines the equilibria of the Volterra–Lotka equations describing prey–predator interactions with diffusion. In this paper, in order to improve the efficiency of the method, we show how Newton's method can be successfully combined with the previous scheme to greatly accelerate the convergence. In some particularly ‘difficult’ problems, the new method reduces the average number of iterations necessary to generate each element of the monotone sequences from 15 to about 3.

Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations

Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations

Resonance, distributions and semilinear elliptic partial differential equations

Soit Ω un domaine borne de R N et Lu=∑ | α | β |≤m (−1) | β | Dβ(aαβ(x)Dαu) un operateur differentiel agissant sur des fonctions a valeurs reelles u(x) definies dans Ω satisfaisant les conditions suivantes: 1) L est uniformement elliptique; 2) les coefficients aαβ sont des fonctions a valeurs reelles dans L ∞ (Ω) et les coefficients d'ordre eleve aαβ, |α|=|β|=m ne sont des fonctions uniformement continues dans Ω; L est symetrique. On obtient un resultat de resonance aux valeurs propres d'ordre eleve