Regularity results for solutions to elliptic obstacle problems in limit cases

We prove the Lewy–Stampacchia’s inequality for elliptic variational inequalities with obstacle involving Leray–Lions type operator whose simpler model case is given by the following \(\begin{aligned} u \in W^{1,N}_0(\Omega )\mapsto -\Delta _N u-\text {div}\left( B (x) |u|^{N-2}u \right) \end{aligned}\) $$\begin{array}{c}\hfill u\in W_0^{1,N}(\mathrm{\Omega })\mapsto -{\mathrm{\Delta }}_{N}u-\text{div}\left({B(x)|u|}^{N-2},u\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a smooth bounded domain of \(\mathbb {R}^N\) ${\mathbb{R}}^N$ with \(N\geqslant 2\) $N⩾2$ , \(\Delta _N u\) ${\mathrm{\Delta }}_{N}u$ denotes the classical N–Laplacian operator and the coefficient \(B:\Omega \rightarrow \mathbb {R}^N\) $B:\mathrm{\Omega }\to {\mathbb{R}}^N$ belongs to a suitable Lorentz–Zygmund space. For this kind of obstacle problems, we also provide regularity results and amongst them we give sufficient conditions to get boundedness of solutions.