HARDY-SOBOLEV-MAZ'YA INEQUALITIES: SYMMETRY AND BREAKING SYMMETRY OF EXTREMAL FUNCTIONS

摘要

Denote points in R-k x RN-k as pairs xi = (x, y), and assume 2 <= k < N. In this paper, we study the problem {-Delta upsilon = lambda vertical bar x vertical bar(-2) upsilon + vertical bar x vertical bar(-b) upsilon(p-1) in R-N, x not equal 0 upsilon > 0, where p > 2, b = N - pN-2/2 and lambda <= (k-2/2)(2), the Hardy constant. Our results are the following: (i) Let p < 2(N-k+1)*. Then there exists at least an entire cylindrically symmetric solution. (ii) Let p <= 2(N)* and lambda >= 0. Then any solution upsilon is an element of L-p(RN; vertical bar x vertical bar(-b) d xi) is cylindrically symmetric. (iii) Let p < 2(N)* and lambda <= (k-2/2)(2) - k-1/p-2. Then ground state solutions are not cylindrically symmetric, and therefore there exist at least two distinct entire solutions. We prove also similar results for the degenerate problem {-div(vertical bar x vertical bar(-2 alpha)del u = vertical bar x vertical bar(-beta p)u(p-1), x not equal 0 u > 0, namely, for the Euler-Lagrange equations of the Maz'ya inequality with cylindrical weights.