Radial Symmetry And Monotonicity For High-order Elliptic Equations

In this paper,we consider the following equationWe prove that every positive solution of the above equation is radially symmetric and monotonous.Furthermore,we consider the following system in RnEvery positive solution of the above equations (0-1) is radially symmetric and monotonous.Our main results are the following theorems:1. Let u(x) be a smooth solution of the equation where ga is the Bessel kernel and*denotes the convolution of function. then the solution satisfies 2.For any r>max{p,g,n(p-1)α,n(q-1)/α),if u∈Lr(Rn) is a positive solution of u(x)=gα*(c(up(z)+uq(x))), where gα is the Bessel kernel and*denotes the convolution of function. then u must be radially symmetric and monotonely decrease about some point.3. Let(u(x),v(x),w(x))be a triple of solutions of(0-1). Assume that u∈Lr(Rn), v∈Lr(Rn),w∈Lr(Rn),where r>max{p,q,n(p1-1)α,n(p2-1)/α,n(p3-1)/α},then(u(X),v(x), w(x))must be radially symmetric and monotonely decrease about some point.To prove the radial symmetry and monotonicity of the solutions,we use the Hardy-Littlewood-Sobolev inequality, the integral form of moving plane method and the properties of the Bessel Potentials etc.