MULTIPLICITY RESULTS FOR A DEGENERATE QUASILINEAR ELLIPTIC EQUATION IN HALF-SPACE

In this work, we prove a multiplicity result for a class of quasilinear elliptic equation involving the subcritical Hardy-Sobolev exponent, and singularities both in the operator and in the non-linearity. Precisely, we study the problem {-div[vertical bar x(N)vertical bar(-ap)vertical bar del u vertical bar(p-2)del u] + lambda vertical bar x(N)vertical bar(-(a+1-c)p)vertical bar u vertical bar(p-2)u = vertical bar x(N)vertical bar-(bp)vertical bar u vertical bar(q-2)u + f in R(+)(N) u = 0 on partial derivative R(+)(N), where we denote x = (x(1), x(2), ..., x(N)) = (x', x(N)) is an element of R(N-1) x R, R(+)(N) = {x is an element of R(N) : x(N) > 0}, partial derivative R(+)(N) = {x is an element of R(N) : x(N) = 0}, and we consider 1 < p < N, 0 <= a (N - p)/p, a < b < a + 1, c = 0, d equivalent to a + 1 - b, q = q(a, b) equivalent to Np/(N - pd) (the Hardy-Soboley critical exponent), lambda is an element of R is a parameter, and f is an element of (L(b)(q)(R(+)(N)))*, the dual space of the weighted Lebesgue space. We prove an existence result for the case f equivalent to 0 and a multiplicity result in the case lambda = 0 for non-autonomous perturbations f not equivalent to 0.