TY - JOUR
A2 - Moller, Manfred H.
AU - Zhidkov, Peter
PY - 2009
DA - 2009/11/19
TI - On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball
SP - 243048
VL - 2009
AB - We consider the following eigenvalue problem: −Δu+f(u)=λu, u=u(x), x∈B={x∈ℝ3:|x|<1}, u(0)=p>0, u||x|=1=0, where p is an arbitrary fixed parameter and f is an odd smooth function. First, we prove that for each integer n≥0 there exists a radially symmetric eigenfunction un which possesses precisely n zeros being regarded as a function of r=|x|∈[0,1). For p>0 sufficiently small, such an eigenfunction is unique for each n. Then, we prove that if p>0 is sufficiently small, then an arbitrary sequence of radial eigenfunctions {un}n=0,1,2,…, where for each n the nth eigenfunction un possesses precisely n zeros in [0,1), is a basis in L2r(B) (L2r(B) is the subspace of L2(B) thatconsists of radial functions from L2(B). In addition, in the latter case, the sequence {un/∥un∥L2(B)}n=0,1,2,… is a Bari basis in the same space.
SN - 0161-1712
UR - https://doi.org/10.1155/2009/243048
DO - 10.1155/2009/243048
JF - International Journal of Mathematics and Mathematical Sciences
PB - Hindawi Publishing Corporation
KW -
ER -