Study On The Behavior Of Rupture Solutions For A Class Of Elliptic MEMS Equations

For elliptic MEMS equation ?u=?|x|~?/u~p,x?R~2\{0}(0.2)where ?>0,?>-2,p>0,this paper studies the behavior of its nonnegative rupture solutions with u(0)=0 in R2.Firstly,the stationary equations are classified according to whether they only have a trivial solution for different(?,p).And it is proved that each solution of(0.2)converges to a corresponding solution of the stationary equation at the origin or infinity.Therefore we classify singularity of the rupture solutions according to(?,p):if the solution converges to the trivial solution at the origin(resp.infinity),then it is called asymptotically isotropic at the origin(resp.infinity).Otherwise it is called asymptotically anisotropic.Then,we construct an energy functional such that the energy of the solution at the origin is less than or equal to that at infinity.By studying the monotonicity of the value to the stationary solution's energy with respect to its period,necessary conditions for the global solution connecting the origin and infinity are obtained.In particular,it's proved that for p=3,the energy is independent of the period.It means that the global solution can only connect two stationary solutions with the same period.Finally,the second order approximation of the asymptotically isotropic solution at the origin or infinity is given by Fourier analysis.