科研之友
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摘要
Let Omega be a planar domain such that Omega is symmetric with respect to both the x- and y-axes and Omega satisfies certain conditions. Then the second eigenvalue of the Dirichlet Laplacian on Omega, v(2)(Omega), is simple, and the corresponding eigenfunction is odd with respect to the y-axis. Let f is an element of C(3) be a function such that
f'(0) > 0, f'''(0) < 0, f (-u) = -f(u) and d/du (f(u)/u) < 0 for u > 0.
Let C denote the maximal continua consisting of nontrivial solutions, {(lambda, u)}, to
Delta u + lambda f(u) = 0 in Omega, u = 0 on partial derivative Omega
and emanating from the second eigenvalue (v(2)(Omega)/f' (0), 0). We show that, for each (lambda, u) is an element of C, the Morse index of u is one and zero is not an eigenvalue of the linearized problem.
- 出版日期2011-3
