In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is
(P)
{
−
△
p
u
−
div
(
c
(
x
)
|
u
|
γ
)
+
b
(
x
)
|
∇
u
|
λ
=
μ
a
m
p
;
in
Ω
,
u
=
0
a
m
p
;
in
∂
Ω
,
\begin{equation}\tag {P} \begin {cases} - \bigtriangleup _p u -\operatorname {div}(c(x)|u|^{\gamma })+b(x)|\nabla u|^{\lambda } =\mu & \text {in $\Omega $},\\ u=0 & \text {in $\partial \Omega $}, \end{cases} \end{equation}
where
Ω
\Omega
is a bounded open subset of
R
N
\mathbb {R}^N
,
N
≥
2
N\geq 2
,
△
p
\bigtriangleup _p
is the so-called
p
−
p-
Laplace operator,
1
>
p
>
N
1> p> N
,
μ
\mu
is a Radon measure with bounded variation on
Ω
\Omega
,
0
≤
γ
≤
p
−
1
0\le \gamma \le p-1
,
0
≤
λ
≤
p
−
1
0\le \lambda \le p-1
, and
|
c
|
|c|
and
b
b
belong to the Lorentz spaces
L
N
p
−
1
,
r
(
Ω
)
L^{\frac {N}{p-1},r}(\Omega )
,
N
p
−
1
≤
r
≤
+
∞
\frac {N}{p-1}\leq r \leq +\infty
, and
L
N
,
1
(
Ω
)
L^{N,1}(\Omega )
, respectively. In particular we prove the existence under the assumptions that
γ
=
λ
=
p
−
1
\gamma =\lambda =p-1
,
|
c
|
|c|
belongs to the Lorentz space
L
N
p
−
1
,
r
(
Ω
)
L^{\frac {N}{p-1},r}(\Omega )
,
N
p
−
1
≤
r
>
+
∞
\frac {N}{p-1}\leq r>+\infty
, and
‖
c
‖
L
N
p
−
1
,
r
(
Ω
)
\|c\|_{ L^{\frac {N}{p-1},r}(\Omega )}
is small enough.