I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2052565/existence-of-a-minimizer-of-a-functional-involving-a-power-q) but did not get any answers so I am trying here.
I have the functional $$\int_D|\nabla(u(x))|^{3/2}+|u(x)|^qdx$$
and I want to know if it exists a minimizer satisfying a boundary condition on some bounded $C^1$ domain. I think that the answer depends on $q$ and that one could probably use the the direct method in the calculus of variations.
For which $q$ does it exist a minimizer (in some appropriate Sobolev space)?
