$L_q$ estimation on the least energy solutions

Let us consider the Neumann problem for a quasilinear equation $$ (I_\varepsilon) {\varepsilon^m div($\mid$\nabla_u$\mid$^{m-2}\nabla_u) - u$\mid$u$\mid$^{m-2} + f(u) = 0 in \Omega {\frac{\partial\nu}{\partial u} = 0 on \partial\Omega. $$ where $1 < m < N, N \geq 2, \varepsilon > 0, \Omega$ is a smooth bounded domain in $R^n$ and $\nu$ is the unit outer normal vector to $\partial\Omega$.