科研之友
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摘要
We consider the nonlinear and nonlocal problem
A(1/2)u = vertical bar u vertical bar(2#)-2(u) in Omega u = 0 on phi Omega
where A(1/2) represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, Omega is a bounded smooth domain in R(n), n >= 2 and 2(#) = 2n/(n - 1) is the critical trace-Sobolev exponent. We assume that Omega is annular-shaped, i.e., there exist R(2) > R(1) > 0 constants such that {x is an element of R(n) s.t. R(1) < vertical bar x vertical bar < R(2)} subset of Omega and 0 is not an element of Q, and invariant under a group Gamma of orthogonal transformations of R(n) without fixed points. We establish the existence of positive and multiple sign changing solutions in the two following cases: if R(1)/R(2) is arbitrary and the minimal Gamma-orbit of Omega is large enough, or if R(1)/R(2) is small enough and Gamma is arbitrary.
- 出版日期2011-11
