Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

Let n >= 3, 0 <= m<n-2/n, rho(1) > 0, beta>beta((m))(0) = ((m(rho 1))/n-2-nm, alpha(m)=((2 beta+rho(1)) (1-m)) and alpha=2 beta+ rho(1). For any lambda>0, we prove the uniqueness of radially symmetric solution v((m)) of (v(m) / of Delta(v(m)/m) +alpha(m)v + beta(x.)del(v) = 0, v >0, in R-n \0 which satisifes solution lim vertical bar x vertical bar -> 0 vertical bar x vertical bar(alpha m) /beta v((m)) (x) = lambda(-) ((rho(1))/((1-m)(beta))) and and obtain higher order estimates of v((m)) near the blow-up point x = 0. We prove that as m -> 0(+) v((m)) converges uniformly in C-2(K) for any compact subset K of R-n \ {0} to the solution v of Delta log v + av + beta x .del v=0 v> 0 in R-n \{0}, which satisfies lim(vertical bar x vertical bar -> 0)vertical bar x vertical bar(alpha/beta)v(x) = lambda(-) (rho 1/beta). We also prove that if the solution u((m)) of u(t) = Delta(u(m)/m), u > 0 in (R-n \{0})x(0,T) whichblows up near {0} x (0, T) at the rate vertical bar x vertical bar(-alpha m/beta) satisfies some mild growth condition on (R-n\{0}) x (0,T), then as m -> 0(+), u((m)) converges uniformly in C-2+theta,C-1+theta/2(K) for some constant theta is an element of (0, 1) and any compact subset K of (R-n \ {0}) x (0,T) to the solution of u(t) = Delta logu, u > 0, in (R-n \ {O}) x (0, T). As a consequence of the proof, we obtain existence of a unique radially symmetric solution v((0)) of Delta log v + alpha v + beta x. del v = 0, v > 0, in R-n \ {0}, which satisfies lim vertical bar x vertical bar -> 0 vertical bar x vertical bar(alpha/beta)v(x)= lambda(-) rho(1)/beta.