Consider the Dirichlet problem
with , , and assume
The existence of solutions to (1) is obtained from conditions on the eigenvalues of the problem
with and . It is first proved that for positive weights , the problem (3) has a sequence of eigenvalues which depend monotonically on the weight. The existence of a solution to
is obtained assuming
where is positive and . Similarly, the existence of a solution to (1) follows assuming (2), where and are positive and for some , . These results are based on degree arguments. In a last section, best Sobolev constants are obtained which imply estimates on the first eigenvalue of (3). These are used in examples.