Semilinear Elliptic Equations Research Articles

Two‐grid weak Galerkin method for semilinear elliptic differential equations

In this paper, we investigate a two‐grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semilinear elliptic equation on the coarse mesh with mesh size , then, we use the coarse mesh solution as an initial guess to linearize the semilinear equation on the fine mesh, that is, on the fine mesh (with mesh size ), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution has sufficient regularity and , the two‐grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

Monotonicity of solutions for fractional p-equations with a gradient term

Abstract In this paper, we consider the following fractional p p -equation with a gradient term: ( − Δ ) p s u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) . {\left(-\Delta )}_{p}^{s}u\left(x)=f\left(x,u\left(x),\nabla u\left(x)). We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional p p -laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, we introduce some new techniques which may also be applied to investigate the qualitative properties of solutions for many problems with gradient terms. Our results are extensions of Berestycki and Nirenberg [Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys. 5 (1988), 237–275] and Wu and Chen [The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), 106933].

Boundary Lipschitz regularity of solutions for general semilinear elliptic equations in divergence form

In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.

Ground state for relativistic nonlinear Schrödinger equations involving general nonlinear term

In this paper, the existence and nonexistence of positive solutions for a relativistic nonlinear Schrödinger equation with vanish potential and general nonlinear term were obtained. First, we reduce this equation to a semilinear elliptic equation by a change. Then, using mountain pass theorem and Strauss compactness lemma, we get the existence of positive radial ground state solutions for this equation. Moreover, the nonexistence of nontrivial solutions for this equation was gotten by Pohozaev identity.

Continuation of solutions to elliptic and parabolic equations on hyperplanes and application to inverse source problems

We consider continuation of solutions u to elliptic and parabolic equations limited on hyperplane under some symmetry conditions of the coefficients: for two domains γ and Γ on a hyperplane in satisfying γ ⊂ ⊂ Γ, we prove conditional stability estimates of u|Γ by u| γ for an elliptic equation and u|Γ×I by u| γ×(0,T) for a parabolic equation with open interval I ⊂ ⊂ (0, T). The proof is based on the even extension and conditional stability for Cauchy problems for elliptic and parabolic equations. We apply the result to prove the uniqueness for an inverse source problem for a heat equation of determining a spatial factor on Γ only by data on γ. Furthermore we provide characterizations of hyperplanes admitting such continuation. Finally we discuss such continuation for semi-linear elliptic equations.

Global structure of steady-states to the full cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model

In a previous paper [10], the author studied the asymptotic behavior of coexistence steady-states to the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. As a result, he proved that the asymptotic behavior can be characterized by a limiting system that consists of a semilinear elliptic equation and an integral constraint. This paper studies the set of solutions of the limiting system. The first main result gives sufficient conditions for the existence/nonexistence of nonconstant solutions to the limiting system by a topological approach using the Leray-Schauder degree. The second main result exhibits a bifurcation diagram of nonconstant solutions to the one-dimensional limiting system by analysis of a weighted time-map and a nonlocal constraint.

A probabilistic approach to Neumann problems for elliptic PDEs with nonlinear divergence terms

By a probabilistic method, we prove the existence and uniqueness of weak solutions to Neumann problems for a class of semi-linear elliptic partial differential equations with nonlinear singular divergence terms, which can only be understood in distributional sense. This leads to the further study on a new class of infinite horizon backward stochastic differential equations, which involves integrals with respect to a forward–backward martingale and a singular continuous increasing process.

Local and global behavior of solutions to 2 D-elliptic equation with exponentially-dominated nonlinearities

We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

Error Estimates for a Pointwise Tracking Optimal Control Problem of a Semilinear Elliptic Equation

We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and obtain first order optimality conditions. We also obtain necessary and sufficient second order optimality conditions. To approximate the solution of the aforementioned optimal control problem we devise a finite element technique that approximates the solution to the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. We analyze convergence properties and prove that the error approximation of the control variable converges with rate $\mathcal{O}(h|\log h|)$ when measured in the $L^2$-norm.

Regularity and symmetry for semilinear elliptic equations in bounded domains

In this paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are stable or more generally of finite Morse index or even more generally locally stable.

On some singular semilinear elliptic equation with doubly exponential growth at infinity in $${\mathbb {R}}^2 $$

On some singular semilinear elliptic equation with doubly exponential growth at infinity in $${\mathbb {R}}^2 $$

Existence of solutions for nonlinear elliptic equations modeling the steady flow of the antarctic circumpolar current

In this paper, we study the steady and purely azimuthal flow model representing the Antarctic Circumpolar Current, which is described as a semi-linear elliptic equation. By means of the Mercator projection, the mathematical model is transferred into a second order elliptic equation. We give sufficient conditions to guarantee the existence of solutions for this second elliptic equation with two forms of nonlinear ocean vorticity term.

Positive Solutions for Slightly Subcritical Elliptic Problems Via Orlicz Spaces

This paper concerns semilinear elliptic equations involving sign-changing weight function and a nonlinearity of subcritical nature understood in a generalized sense. Using an Orlicz–Sobolev space setting, we consider superlinear nonlinearities which do not have a polynomial growth, and state sufficient conditions guaranteeing the Palais–Smale condition. We study the existence of a bifurcated branch of classical positive solutions, containing a turning point, and providing multiplicity of solutions.

Limiting profile of solutions for sublinear elliptic equations with shrinking self-focusing core

ABSTRACT This paper is concerned with the following problem: where , and the self-focusing core supp shrinks to a point as . We show that there is a least energy solution corresponding to and then investigate the limiting profile of concentration for as . The results are supplements to the works of [Ackermann and Szulkin. A concentration phenomenon for semilinear elliptic equations. Arch Ration Mech Anal. 2013;207(3):1075–1089; Fang and Wang. Limiting profile of solutions for Schrödinger equations with shrinking self-focusing core. Calc Var Partial Differential Equations. 2020;59(4):Paper No. 129, 18].

Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

In this paper, we study the following fractional semilinear Neumann problem arising from Keller–Segel model: where is a smooth bounded domain and is the outward unit normal to . First, under the superlinear and subcritical growth assumptions on , we prove that there exists at least one positive nonconstant solution for small and the family of solutions is uniformly bounded. Moreover, we build a Pohozaev‐type identity for the ( ‐) Neumann harmonic extension of the following problem: where is a function such that . As a direct application of this identity, when satisfies , where , we deduce the nonexistence of weak bounded solution in star‐shaped domains.

Large and very singular solutions to semilinear elliptic equations

We consider equation $$-\Delta u+f(x,u)=0$$ in smooth bounded domain $$\Omega \in \mathbb {R}^N$$ , $$N\geqslant 2$$ , with $$f(x,r)>0$$ in $$\Omega \times \mathbb {R}^1_+$$ and $$f(x,r)=0$$ on $$\partial \Omega $$ . We find the condition on the order of degeneracy of f(x, r) near $$\partial \Omega $$ , which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on $$\partial \Omega $$ . Moreover, we prove that the mentioned condition is a sufficient condition for the uniqueness of a large solution and conjecture that this condition is also a necessary condition of the uniqueness.

The semilinear elliptic equations with double weighted critical exponents

In this paper, we focus on the equation with double weighted critical exponents, including Hardy–Sobolev, Sobolev, and Hénon–Sobolev critical exponents. Applying the Nehari manifold and the mountain pass theorem, we prove the existence of the ground state solutions for the equation in RN. Based on this fact, the positive solutions of the equation on the unit ball in RN are established.

Some symmetry properties of gyre flows

The large gyre systems of near-surface almost-circular ocean currents formed by global wind patterns and the effect of the Earth’s rotation often present symmetry properties. By means of maximum principles, we infer certain symmetry properties from the specific structure of such flows. Our approach uses the stereographic projection, which allows us to turn the governing equations for steady flows on a rotating sphere into semilinear elliptic equations in the plane. Furthermore, we discuss the possibility of using conformal mappings to study specific flows.

Existence of solutions for a nonlocal reaction-diffusion equation in biomedical applications

The paper is devoted to a nonlocal semi-linear elliptic equation in ℝn arising in various biological and biomedical applications. The Fredholm property studied for the corresponding linear elliptic operators with discontinuous coefficients allows the application of the implicit function theorem to prove the persistence of solutions under a small perturbation of the problem. Furthermore, the existence of solutions is established by the Leray—Schauder method based on the topological degree for Fredholm and proper operators and on a priori estimates of solutions in some special weighted spaces.