ON SINGULAR SUPERLINEAR ELLIPTIC PROBLEMS OF 2ND AND 4TH ORDERS

We study the positive radially symmetric solutions of the homogeneous Dirichlet problem DELTA(m) u = (- 1)m a(Absolute value of x)(1 - Absolute value of x)-lambda u(beta) in B, u = ... = (partial derivative/partial derivative v)m-1 u = 0 on partial derivative B, where lambda > 0, beta > 1 are constants, B denotes the unit ball centered at the origin in R(n), partial derivative B is the boundary of B, partial derivative/partial derivative v is the outward normal derivative and the function a greater-than-or-equal-to 0 is continuous in [0,1], locally Holder-continuous in [0,1) with a(1) not-equal 0. We deal with the cases m = 1,2 and n greater-than-or-equal-to 2m + 1. Our main tool is the well known "Mountain Pass Theorem". To verify the hypotheses of the theorem we need some compactness results.