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Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-\\Delta u=g(u)$ in a bounded domain $\\Omega$ of $\\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(\\Omega)$ bound for the extremal solution of $-\\Delta u=\\lambda f(u)$ when $n\\geq 5$ and the domain is convex. 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