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Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain and $f\in L^q$ with $q\in (1,\infty)$. If $u\in H_0^1(\Omega)$ satisfies $$\int_\Omega \nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in H_0^1(\Omega)$$

then we can conclude that there exist some constant $C>0$ such that $$\|u\|_{1,q}\leq C(\|u\|_q+\|f\|_q)$$

Now suppose that $p\in (1,\infty)$ and $u\in W_0^{1,p}(\Omega)$ satisfies $$\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in W_0^{1,p}(\Omega)$$

Can we conclude that there exist some constant $C>0$ such that $$\|u\|_{1,q}\leq C(\|u\|_q+\|f\|_q)$$

Notes: The constant $C$ does not depend on $q$ and $u$. For the case $p=2$, we can show that in fact $u$ is a Strong solution i.e. $-\Delta u=f$ almost everywhere. From this fact we can conclude the first inequality (see for example Gilbard-Trudinger in the section of $L^p(\Omega)$ estimates Chapter 9).

For $p\neq 2$ I dont know if $u$ still being a Strong solution and if the same techinique can be applied.

Any hint or reference would be appreciated.

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up vote 2 down vote accepted

Such estimates are available. It seems that the first such result was Theorem 2 in Projections onto gradient fields and $L^p$ estimates for degenerate elliptic equations by Iwaniec. For a more general form see Theorems 1.1 and 1.2 in "On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems" by DiBenedetto and Manfredi (no free link, sadly - it's in the American Journal of Mathematics). Even more general version, with VMO coefficients, is Theorem 1.4 in A local estimate for nonlinear equations with discontinuous coefficients by Kinnunen and Zhou.

Incidentally, for the latter paper the author-hosted preprint is much more readable than the poorly scanned copy sold by the (expensive) publisher Taylor&Francis.

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