Radially Symmetric Solutions of Δw+wp−1w=0

الملخص EN

We investigate solutions of w′′+N−1/rw′+wp−1w=0, r>0 and focus on the regime N>2 and p>N/(N-2).

Our advance is to develop a technique to efficiently classify the behavior of solutions on (rmin,rmax), their maximal positive interval of existence.

Our approach is to transform the nonautonomous w equation into an autonomous ODE.

This reduces the problem to analyzing the phase plane of the autonomous equation.

We prove the existence of new families of solutions of the w equation and describe their asymptotic behavior.

In the subcritical case N/(N-2)

Our advance is to prove the existence of a family of solutions of the subcritical case which satisfiesw(ri)=w1(ri) for infinitely many values ri>0.

At the critical value p=(N+2)/(N-2) there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions.

In the supercritical regime p>(N+2)/(N-2) we prove the existence of a family of “super singular” sign changing singular solutions.