AN EIGENVALUE PROBLEM RELATED TO THE CRITICAL SOBOLEV EXPONENT: VARIABLE COEFFICIENT CASE

作者：Takahashi Futoshi^*

来源：Differential and Integral Equations, 2010, 23(9-10): 827-860.

Let u(epsilon) be a least energy solution to the nearly critical problem
-Delta u = c(0)K(x)u(p epsilon) in Omega, u > 0 in Omega, u vertical bar(partial derivative Omega) = 0,
where Omega subset of R(N) (N >= 3) is a smooth bounded domain, c(0) = N(N - 2), p(epsilon) = (N + 2)/(N - 2) - epsilon where epsilon > 0 is a small parameter and K is an element of C(2)((Omega) over bar) is a positive function. Under some assumptions on K, we prove several asymptotic estimates of the eigenvalues lambda(i,epsilon) and corresponding eigenfunctions v(i,epsilon) to the eigenvalue problem
{Delta v(i,epsilon) = lambda(i,epsilon) (c(0)p(epsilon) K(x)u(epsilon)(p epsilon - 1))v(i,epsilon) in Omega, v(i,epsilon) = 0 on partial derivative Omega, parallel to v(i,epsilon)parallel to L(infinity)(Omega) = 1
as epsilon -> 0, for i = 2, ..., N + 1, N +2.

• 出版日期2010-10