Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

@article{Cao2019ClassificationON,
  title={Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity},
  author={Daomin Cao and Wei Dai},
  journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  year={2019},
  volume={149},
  pages={979 - 994}
}
  • D. Cao, Wei Dai
  • Published 1 August 2019
  • Mathematics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Abstract In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 < γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non… 

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References

SHOWING 1-10 OF 36 REFERENCES

Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation

AbstractIn this paper, we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary Choquard equation $$\Delta

A classification of solutions of a conformally invariant fourth order equation in Rn

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Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. We use the method of moving planes in integral forms

Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space

AbstractIn this paper, let α be any real number between 0 and 2, we study the Dirichlet problem for semi-linear elliptic system involving the fractional Laplacian:

A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

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Classification of solutions for an integral equation

Let n be a positive integer and let 0 < α < n. Consider the integral equation $$(0.1) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; u(x) = \int\limits^{}_{R^{n}} {1 \over |x -y|^{n-\alpha}}u(y)^{{n+\alpha} \over

Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1 <p, q, n/X < x. When p =q' or p = 2

On uniqueness of solutions of $n$-th order differential equations in conformal geometry

In this paper, we prove a uniqueness theorem for an n-th order elliptic equation on the standard n-sphere S. The problem arises naturally from the point of view of conformal geometry. The method we