Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations

Guo, Yuxia; Peng, Shaolong

doi:10.1007/s00033-021-01551-5

Cited by 16 publications

(5 citation statements)

References 38 publications

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“…then the integrals are well defined by virtue of the asymptotic properties of K ν (r) when r → 0 and r → ∞, see [13,18]. The qualitative properties of solutions to different equations has been taken into account by many researchers, for example, see [2,14,17].…”

Section: Introductionmentioning

confidence: 99%

“…(1.7) into the equivalent system involving both the pseudo-relativistic operator (−∆ + m 2 ) s and the fractional Laplacian (−∆) t . For the transformed system, they derived the radial symmetry property and the monotone decreasing property of the nonnegative solution to the system (see [18]). With the information we have, there are few people concerned with qualitative properties of solutions for systems involving the general pseudo-relativistic operators.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems

Chen¹,

Li²,

Bao³

2022

CPAA

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In this paper, we focus on a class of general pseudo-relativistic systems<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \begin{aligned} &(-\Delta+m^2)^su(x) = f(u(x), v(x)), \\ &(-\Delta+m^2)^tv(x) = g(u(x), v(x)), \end{aligned} \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ m \in (0, +\infty) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ s, t \in (0,1) $\end{document}</tex-math></inline-formula>. Before giving the main results, we first introduce a decay at infinity and a narrow region principle. Then we implement the direct method of moving planes to show the radial symmetry and monotonicity of positive solutions for the above system in both the unit ball and the whole space.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems

Chen¹,

Li²,

Bao³

2022

CPAA

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“…Recall that the pseudo-relativistic operators in is defined by (see [1, 7]) where , stands for Cauchy principal value, and denotes the modified Bessel function with order , which solves the equation and satisfies the following integral representation It is easy to verify that is a real and positive function satisfying for all , for . Moreover, for , it holds that (see [26–28, 31]) Hence there exists a small and two constants such that Next, for , we can also derive that Hence there exists a large and two constants such that Let Then it is easy to verify that for any , the integral on the right-hand side of the definition (1.4) is well-defined. Hence makes sense for all functions .…”

Section: Introductionmentioning

confidence: 99%

“…It is easy to verify that K ν (r) is a real and positive function satisfying K ν (r) < 0 for all r > 0, K ν = K −ν for ν < 0. Moreover, for ν > 0, it holds that (see [26][27][28]31])…”

Section: Introductionmentioning

confidence: 99%

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

Guo¹,

Peng²

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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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.

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“…And pseudo-relativistic Hartree equation can describe an N -body quantum system of relativistic bosons with two-body interaction given by Newtonian gravity (see [17,18]). We refer to [1,9,14,19] for the rigorous derivation of the equations and the study of their dynamical properties. The solution u to (1.1) is also a ground state or a stationary solution to the following Schrödinger equation with nonlinearities:…”

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confidence: 99%

Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity

Guo

Peng

2022

CPAA

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In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\Delta+m^{2})^{s}u = a(x_1)f\left(u,\nabla u\right),\quad {\rm{in}} \,\,\mathbb R^{N}, $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula>, mass <inline-formula><tex-math id="M2">\begin{document}$ m>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> is a non-decreasing functions. We prove the nonexistence and the monotonicity of the positive bounded solution for the above equation via the direct method of moving planes.

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Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations

Cited by 16 publications

References 38 publications

Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems

Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems

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

Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity

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