Semilinear Elliptic Systems Research Articles

Liouville type theorems involving fractional order systems

Abstract In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).

Constant sign and nodal solutions for singular Gierer–Meinhardt-type system

We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray–Schauder topological degree involving perturbation argument.

The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core

The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core

Blow-up solutions for a 4-dimensional semilinear elliptic system of Liouville type in some general cases

This paper is devoted to the existence of singular limit solutions for a nonlinear elliptic system of Liouville type under Navier boundary conditions in a bounded open domain of R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{4}$\\end{document}. The concerned results are obtained employing the nonlinear domain decomposition method and a Pohozaev-type identity.

Existence of non-radial solutions to semilinear elliptic systems on a unit ball in $${\mathbb {R}}^3$$

Existence of non-radial solutions to semilinear elliptic systems on a unit ball in $${\mathbb {R}}^3$$

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

A system of superlinear elliptic equations in a cylinder

The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Omega =Omega 'times (0,a)subset {mathbb {R}}^n, where Omega 'subset {mathbb {R}}^{n-1} is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Omega with another superlinear (or linear) equation defined at the bottom of the cylinder Omega 'times {0}. Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L^infty bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.

Multiplicity of Positive Solutions for a Semilinear Elliptic System with Strongly Coupled Critical Terms and Concave Nonlinearities

Multiplicity of Positive Solutions for a Semilinear Elliptic System with Strongly Coupled Critical Terms and Concave Nonlinearities

Isolated singularities for semilinear elliptic systems with Hardy potential

In this paper, we analyze the asymptotic behavior near zero of positive solutions for a class of semilinear elliptic systems with Hardy-type singular potential. In addition, we address the questions of existence and uniqueness for the solutions.

Global bifurcation of positive solutions for a class of superlinear elliptic systems

We are concerned with the global bifurcation of positive solutions for semilinear elliptic systems of the form { − Δ u = λ f ( u , v ) in Ω , − Δ v = λ g ( u , v ) in Ω , u = v = 0 on ∂ Ω , where λ ∈ R is the bifurcation parameter, Ω ⊂ R N , N ≥ 2 is a bounded domain with smooth boundary ∂ Ω . We establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ -direction. The nonlinearities f , g ∈ C 1 ( R × R , ( 0 , ∞ ) ) are nondecreasing for each variable and have superlinear growth at infinity. The proof of our main result is based upon bifurcation theory. In addition, as an application for our main result, when f and g subject to the upper growth bound, by a technique of taking superior limit for components, then we may show that the branch must bifurcate from infinity at λ = 0 .

Nodal solutions for singular semilinear elliptic systems

In this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with perturbation arguments involving singular terms.

Positive solutions for a class of concave-convex semilinear elliptic systems with double critical exponents

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents: { − Δ u = | u | 2 ∗ − 2 u + α 2 ∗ | u | α − 2 | v | β u + λ | u | q − 2 u , in Ω , − Δ v = | v | 2 ∗ − 2 v + β 2 ∗ | u | α | v | β − 2 v + μ | v | q − 2 v , in Ω , u , v > 0 , in Ω , u = v = 0 , on ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary,~ λ , μ > 0 , 1 < q < 2 , α > 1 , β > 1 , α + β = 2 ∗ = 2 N N − 2 . By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature.

Qualitative properties of blow-up solutions to some semilinear elliptic systems in non-convex domains

In this paper we study qualitative properties of boundary blow-up solutions to some semilinear elliptic cooperative systems in bounded non-convex domains. In particular, by a careful adaptation of the celebrated moving plane procedure of Alexandrov–Serrin, we deduce symmetry and monotonicity results for blow-up solutions for this class of systems.

Existence of normalized solutions for semilinear elliptic systems with potential

In this paper, we consider the existence of normalized solutions to the following system: −Δu + V1(x)u + λu = μ1u3 + βv2u and −Δv + V2(x)v + λv = μ2v3 + βu2v in R3, under the mass constraint ∫R3u2+v2=ρ2, where ρ is prescribed, μi, β > 0 (i = 1, 2), and λ∈R appears as a Lagrange multiplier. Then, by a min–max argument, we show the existence of fully nontrivial normalized solutions under various conditions on the potential Vi:R3→R(i=1,2).

Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.

Positive solutions for semilinear elliptic systems with boundary measure data

Positive solutions for semilinear elliptic systems with boundary measure data

Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ -\Delta v+\lambda _2v=\mu _2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad \int _{{{\mathbb {R}}^N}}v^2=a_2^2, \end{array}\right. } \end{aligned}$$where \(p,r_1+r_2<2^*\) and \(q\le 2^*\). To this purpose, we study the geometry of the Pohozaev manifold and the associated minimization problem. Under some assumptions on \(a_1,a_2\) and \(\beta \), we obtain the existence of the positive normalized ground state solution to the above system.

A nondegeneracy condition for a semilinear elliptic system and the existence of multibump solutions

A nondegeneracy condition for a semilinear elliptic system and the existence of multibump solutions

Structure of sets of solutions of parametrised semi-linear elliptic systems on spheres

In this paper we study a parametrised non-cooperative symmetric semi-linear elliptic system on a sphere. Assuming that there exist critical orbits of the potential, we study the structure of the sets of solutions of the system. In particular, using the equivariant Rabinowitz alternative we formulate sufficient conditions for a bifurcation of unbounded sets of solutions.

Asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property

We prove an asymptotic monotonicity formula for bounded, globally minimizing solutions (in the sense of Morse) to a class of semilinear elliptic systems of the form \(\Delta u= W_u(u)\), \(x\in \mathbb{R}^n\), \(n\geq 2\), with \(W:\mathbb{R}^m\to \mathbb{R}\), \(m\geq 1\), nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution \(u\)). As an application, we can prove a Liouville type theorem under various assumptions. For more information see https://ejde.math.txstate.edu/Volumes/2021/04/abstr.html