# 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} }

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

- Mathematics
- 2010

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

- Mathematics
- 1998

Abstract. In this paper, we consider the following conformally invariant equations of fourth order¶
$ \cases {\Delta^2 u = 6 e^{4u} &in $\bf {R}^4,$ \cr e^{4u} \in L^1(\bf {R}^4),\cr}$(1)¶and¶
$…

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

- Mathematics
- 2017

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

- Mathematics
- 2017

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:…

### Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

- Mathematics
- 2013

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

- Mathematics
- 2015

Abstract In this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝn+: where for i = 1, 2. First, we show that, under some mild growth…

### Classification of solutions for an integral equation

- Mathematics
- 2006

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

- Mathematics
- 1983

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

- Mathematics
- 1997

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…