Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy-Leray inequalities

The purpose of this paper is twofold. First is the study of the nonexistence of positive solutions of the parabolic problem {partial derivative u/partial derivative t = Delta(p)u + V(x)u(p-1) + lambda u(q) in Omega x (0, T), u(x, 0) = u(0)(x) >= 0 in Omega, vertical bar del u vertical bar(p-2)partial derivative u/partial derivative v = beta vertical bar u vertical bar(p-2)u on partial derivative Omega x (0, T), where Omega is a bounded domain in R-N with smooth boundary partial derivative Omega, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u) is the p-Laplacian of u, V is an element of L-l(oc)1 (Omega), beta is an element of L-loc(1)(partial derivative Omega), lambda is an element of R, the exponents p and q satisfy 1 < p < 2, and q > 0. Then, we present some sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation we are interested in.