Semilinear Elliptic Equations Research Articles

On nonlocal systems with jump processes of finite range and with decays

We study the following system of equationsLi(ui)=Hi(u1,⋯,um)inRn, when m≥1, ui:Rn→R and H=(Hi)i=1m is a sequence of general nonlinearities. The nonlocal operator Li is given byLi(f(x)):=limϵ→0⁡∫Rn∖Bϵ(x)[f(x)−f(z)]Ji(z−x)dz, for a sequence of even, nonnegative and measurable jump kernels Ji. We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel Ji. In particular, for the case of scalar equations, that is when m=1, it reads∬R2nAy(∇xu)[η2(x)+η2(x+y)]J(y)dxdy≤∬R2nBy(∇xu)[η(x)−η(x+y)]2J(y)dxdy, for any η∈Cc1(Rn) and for some nonnegative Ay(∇xu) and By(∇xu). This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in [46] for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever Hi(u)≥0 or ∑i=1muiHi(u)≤0 then Liouville theorems hold for each ui in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel Ji and a Liouville theorem for the quotient of partial derivatives of u.

On an iterative process for the grid conjugation problem with iterations on the boundary of the solution discontinuity

An iterative process for the grid problem of conjugation with iterations on the boundary of the discontinuity of the solution is considered. Similar grid problem arises in difference approximation of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions. The study of iterative processes for the states of such problems is of independent interest for theory and practice. The paper shows that the numerical solution of boundary problems of this type can be efficiently implemented using iterations on the inner boundary of the grid solution discontinuity in combination with other iterative methods for nonlinearities separately in each of the grid subregions. It can be noted that problems for states of controlled processes described by equations of mathematical physics with discontinuous coefficients and solutions arise in mathematical modeling and optimization of heat transfer, diffusion, filtration, elasticity theory, etc. The proposed iterative process reduces the solution of the initial grid boundary problem for a state with a discontinuous solution to a solution of two special boundary problems in two grid subdomains at every fixed iteration. The convergence of the iteration process in the Sobolev grid norms to the unique solution of the grid problem for each initial approximation is proved.

A note on steady vortex flows in two dimensions

In this note, we give a general criterion for steady vortex flows in a planar bounded domain. More specifically, we show that if the stream function satisfies “locally” a semilinear elliptic equation with monotone or Lipschitz nonlinearity, then the flow must be steady.

Asymptotic behavior and existence of solutions for singular elliptic equations

We study the asymptotic behavior, as $$\gamma $$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated with singular semilinear elliptic equations whose model is $$\begin{aligned} -\Delta u=\frac{f(x)}{u^\gamma }\,\text { in }\Omega , \end{aligned}$$ where $$\Omega $$ is an open, bounded subset of $${\mathbb {R}}^{N}$$ and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f: either strictly positive on every compactly contained subset of $$\Omega $$ or only nonnegative. Through this study, we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated with $$\begin{aligned} -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text { in }\Omega . \end{aligned}$$

Adaptive fixed point iterations for semilinear elliptic partial differential equations

In this paper we study the behaviour of finite dimensional fixed point iterations, induced by discretization of a continuous fixed point iteration defined within a Banach space setting. We show that the difference between the discrete sequence and its continuous analogue can be bounded in terms depending on the discretization of the infinite dimensional space and the contraction factor, defined by the continuous iteration. Furthermore, we show that the comparison between the finite dimensional and the continuous fixed point iteration naturally paves the way towards a general a posteriori error analysis that can be used within the framework of a fully adaptive solution procedure. In order to demonstrate our approach, we use the Galerkin approximation of singularly perturbed semilinear monotone problems. Our scheme combines the fixed point iteration with an adaptive finite element discretization procedure (based on a robust a posteriori error analysis), thereby leading to a fully adaptive fixed-point-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach.

Uniqueness and local properties of positive solutions to a semilinear elliptic equation with singular potentials

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

Stuart-type vortices modeling the Antarctic Circumpolar Current

Using the method of sub- and supersolutions for semilinear elliptic equations, in combination with the stereographic projection, we show that Stuart-type vortices can model the steady flow of the Antarctic Circumpolar Current.

Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

In this paper, we consider the following semilinear elliptic equation:where $\Omega$ is an exterior domain in $\mathbb{R}^N$ with $N\ge~3$, $h:~\Omega\times~\mathbb{R}^+\to~\mathbb{R}$ is a measurable function, andderive optimal nonexistence results of positive supersolutions. Our argument isbased on a nonexistence result of positive supersolutions of a linear ellipticproblem with Hardy potential. We also establish sharp nonexistence results of positive supersolutionsto an elliptic system.

Classification of radial solutions to Hénon type equation on the hyperbolic space

We devote this paper to classifying radial solutions of a weighted semilinear elliptic equation on the hyperbolic space. More precisely, for a weighted Lane-Emden equation on the hyperbolic space, we shall study the sign and asymptotic behavior of the radial solutions. We shall also show the existence of fast-decay sign-changing solutions to the Lane-Emden equation on the hyperbolic space.

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schr\"odinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms

We consider the structure and the stability of positive radial solutions of a semilinear inhomogeneous elliptic equation with multiple growth terms \begin{document}$ \Delta u+\sum\limits_{i = 1}^{k}K_i(|x|)u^{p_i}+\mu f(|x|) = 0, \quad x\in\mathbb{R}^n, $\end{document} which is a generalization of Matukuma's equation describing the dynamics of a globular cluster of stars. Equations similar to this kind have come up both in geometry and in physics, and have been a subject of extensive studies. Our result shows that any positive radial solution is stable or weakly asymptotically stable with respect to certain norm.

Nonhomogeneous polyharmonic elliptic problems involving GJMS operator on Riemannian manifold

Let [Formula: see text] be a closed Riemannian manifold, under some assumptions on [Formula: see text] and [Formula: see text] by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: [Formula: see text]. When [Formula: see text] is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.

The Dirichlet problem for semilinear elliptic equations in an infinite sector

We consider the Dirichlet problem for a semilinear elliptic equation in an unbounded sectorial domain $$\Omega $$ of the two-dimensional space. The problem is supplemented with a limiting behavior related to a prescribed root z of the nonlinearity of the equation. The one-dimensional setting of the problem has a unique solution $$V_{z}$$, and the goal of the present paper is to construct a two-dimensional positive and bounded solution u approaching $$V_{z}(d)$$ when $$d = d(x,\partial \Omega )\rightarrow \infty $$. This is established using sub- and supersolutions method and employing a sliding argument.

Exact asymptotic behavior of singular positive solutions of fractional semi-linear elliptic equations

In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-\Delta)^\sigma u = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $\sigma \in (0, 1)$ and $\frac{n}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$.

A type of cascadic multigrid method for coupled semilinear elliptic equations

This paper is to introduce a type of cascadic multigrid method for coupled semilinear elliptic equations. Instead of solving the coupled semilinear elliptic equation on a very fine finite element space directly, the new scheme needs to solve a decoupled linear system by some smoothing steps on each of multilevel finite element spaces and solve a coupled semilinear elliptic equation on a coarse space. By choosing the appropriate number of smoothing steps on different finite element spaces, the corresponding optimal convergence rate and optimal computation work can be derived. Besides, the adaptive cascadic multigrid method for coupled semilinear elliptic equations and its analysis are also presented theoretically and numerically. Moreover, the requirement of bounded second order derivatives of the nonlinear term in the existing multigrid methods is reduced to the Lipschitz continuation property in the presented new scheme. Some numerical experiments are presented to validate our theoretical analysis.

Existence of solutions for singular critical semilinear elliptic equation

This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved.

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.

The Existence of Positive Solution for Semilinear Elliptic Equations with Multiple an Inverse Square Potential and Hardy-Sobolev Critical Exponents

Via the concentration compactness principle, delicate energy estimates, the strong maximum principle, and the Mountain Pass lemma, the existence of positive solutions for a nonlinear PDE with multi-singular inverse square potentials and critical Sobolev-Hardy exponent is proved. This result extends several recent results on the topic.

Pointwise estimates of solutions to semilinear elliptic equations and inequalities

We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $\Omega$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.

Interior gradient and Hessian estimates for the Dirichlet problem of semi-linear degenerate elliptic systems: A probabilistic approach

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.