AN EIGENVALUE PROBLEM RELATED TO BLOWING-UP SOLUTIONS FOR A SEMILINEAR ELLIPTIC EQUATION WITH THE CRITICAL SOBOLEV EXPONENT

作者：Takahashi Futoshi^*

来源：Discrete and Continuous Dynamical Systems - Series S, 2011, 4(4): 907-922.

DOI：10.3934/dcdss.2011.4.907

• Critical Sobolev exponent
• blowing-up solutions
• eigenvalue problem

We consider the eigenvalue problem { -Delta v = lambda (copu(epsilon)(p-1) + epsilon) v in Omega,
v = 0 on partial derivative Omega,
vertical bar vertical bar v vertical bar vertical bar L infinity(Omega) = 1 IlvIlL-02) =1
where Q subset of R-N(N >= 5) is a smooth bounded domain, co = N(N - 2), p = (N + 2)/(N - 2) is the critical Sobolev exponent and epsilon > 0 is a small parameter. Here u, is a positive solution of
-Delta u = cou(p) + epsilon u in Omega, u vertical bar partial derivative Omega = 0
with the property that
integral Omega vertical bar del u epsilon vertical bar(2)dx/(integral Omega vertical bar u(epsilon vertical bar)(p+1)dx)2/p+1 -> S-N as epsilon -> 0,
where S-N is the best constant for the Sobolev inequality. In this paper, we show several asymptotic estimates for the eigenvalues lambda(i,epsilon) and corresponding eigenfunctions v(i,epsilon) for i = 1,2, center dot center dot center dot, N + 1, N + 2.

• 出版日期2011-8