STEKLOV EIGENVALUE PROBLEMS WITH INDEFINITE WEIGHT FOR THE (p, q)-LAPLACIAN

This paper provides existence and non-existence results on a positive solution for the problem Delta(r)u+ mu Delta(r')u = |u|(r-2) u+ mu|u|(r'- 2)u, with a nonlinear boundary condition given by <|del u|(r-2)del u+|del u|(r'-2)del u, nu > = lambda m(r)(x)|u|(r-2)u on the boundary of the domain, with mu > 0 and 1 < r not equal r' < infinity, where Omega is a bounded domain in R-N, nu is the outward unit normal vector on partial derivative Omega, <.,.> is the scalar product of R-N and m(r) is a weight function admitting sign-change. We show that existence and non-existence of a positive solution depend only on the relation between lambda and the first eigenvalue of r-Laplacian with weight function m(r), whence it is independent of the operator Delta(r') and the parameter mu > 0.