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We consider a class of Dirichlet boundary problems for nonlinear degenerate elliptic equations with lower order terms. We prove, using symmetrization techniques, pointwise estimates for the rearrangement of the gradient of a solution u and integral estimates. As consequence, we get apriori estimates which show how the summability of the gradient of a solution increases when the summability of the datum increases, also taking into account the presence of a zero order term which can have a regularizing effect.

In this paper, we prove an existence result for weak solutions to a class of Dirichlet boundary value problems whose prototype is \begin{equation*} \label{pa} \left\{ \begin{array}{lll} -\Delta_p u =\beta |\nabla u|^{q} +c(x)|u|^{p-2}u +f & & \text{in}\ \Omega \\ u=0 & & \text{on}\ \partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded open subset of $\mathbb R^N$, $N\geq 2$, $\Delta_p u={\rm div} \left(|\nabla u|^{p-2}\nabla u\right)$, $1 < p < N$, $ p-1 < q\le p-1+\frac p N$, $\beta $ is a positive constant, $c\in L^{\frac N p}(\Omega)$ with $c\ge 0$, $c\neq 0$ and $f\in L^{(p^*)'}(\Omega).$ We further assume smallness assumptions on $c$ and $f$. Our approach is based on Schauder's fixed point theorem.

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is \( - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} \) with Dirichlet boundary condition, where γ is the normalized Gaussian function in \( \mathbb{R}^{n} \). To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.

We prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is u ∈ W 0 1 , p ( Ω ) , - div ( a ( x ) ( 1 + | ∇ u | 2 ) ( p - 2 ) / 2 ∇ u ) + b ( x ) ( 1 + | ∇ u | 2 ) ( σ + 1 ) / 2 = f in D ′ ( Ω ) , where Ω is a bounded open subset of R N , N ⩾ 2 , p satisfies p ⩾ 2 N / ( N + 1 ) , a is a function belonging to L ∞ ( Ω ) such that a ( x ) ⩾ α > 0 , f belongs to the dual space W - 1 , p ′ ( Ω ) , b belongs to some Lebesgue space L r ( Ω ) with r ⩾ r * ( N , p ) and σ belongs to the interval [ 0 , σ * ( N , p , r ) ] , with σ * ( N , p , r ) and r * ( N , p ) functions which are specified below.