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In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type $$\begin{array}{c}f(x,\eta ,\xi )\ge a\left(x\right)\frac{{\left|\xi \right|}^p}{{(1+|\eta \left|\right)}^{\alpha }}-b_1\left(x\right){\left|\eta \right|}^{{\beta }_1}-g_1\left(x\right),\\ f(x,\eta ,0)\le b_2\left(x\right){\left|\eta \right|}^{{\beta }_2}+g_2\left(x\right),\end{array}$$ where$0\le \alpha <p$, $1\le {\beta }_1<p$, $0\le {\beta }_2<p$, $\alpha +{\beta }_i\le p$, $a\left(x\right)$, $b_i\left(x\right)$, $g_i\left(x\right)$ ($i=1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1,p}\left(a\right)$ , which assume a boundary datum $u_0\in W^{1,p}\left(a\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is $$\left\{\begin{array}{cc}-div\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded open subset of ${ℝ}^N$, $N\ge 2$, $2-1/N\&lt;p\&lt;N$, $a$ belongs to $L^{\infty }\left(\Omega \right)$, $a\left(x\right)\ge {\alpha }_0\&gt;0$, $f$ is a function in $L^1\left(\Omega \right)$, $b$ is a function in $L^r\left(\Omega \right)$ and $0\le \lambda \&lt;{\lambda }^{*}(N,p,r),$ for some $r$ and ${\lambda }^{*}(N,p,r)$.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is
$$\left\{-\text{div}\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,\right.$$
where Ω is a bounded open subset of ${ℝ}^N$, N > 2, 2-1/, belongs to  (Ω), $a\left(x\right)\ge {\alpha }_0>0$, is a function in (Ω), is a function in $L^r\left(\Omega \right)$ and 0 ≤ λ < λ *(), for some and λ *().