Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

The Moving Plane Method for Doubly Singular Elliptic Equations Involving a First-Order Term

In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.

Radial symmetry for an elliptic PDE with a free boundary

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u ) in B 1 , 0 ≤ u > M , in B 1 , u = M , on ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

Radial solutions for fully nonlinear elliptic equations of Monge–Ampère type

First, the symmetry of classical solutions to the Monge–Ampère-type equations is obtained by the moving plane method. Then, the existence and nonexistence of radial solutions in a ball are got from the symmetry results. Finally, the existence and nonexistence of classical solutions to Hessian equations in bounded domains are considered.

Qualitative analysis for the nonlinear fractional Hartree type system with nonlocal interaction

In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

Semilinear elliptic equations involving mixed local and nonlocal operators

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ) s , with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation\Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4}to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at {\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution {u(x)=u(|x|)} for any given {u(0)>0}, and the asymptotic behavior of {u(|x|)} at {\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at {\infty}, in turn, determines the radial symmetry of a general positive entire solution {u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at {\infty} is obtained.