A priori estimates for a class of degenerate elliptic equations

Abstract

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega, \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when \({f \in L^q(\Omega), p > 2}\) and q ≥ 2. If u is a weak solution in \({W^{1, p}(\Omega)}\), we obtain estimates for u in the Nikolskii space \({\mathcal{N}^{1+2/r,r}(\Omega)}\), where r = q(p − 2) + 2, in terms of the Lq norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that \({\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}\) for every \({\epsilon > 0}\) sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in \({W^{1,r}(\Omega)}\).