We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. Specifically, we consider problems of the form
\begin{cases} u \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), & u \geq \psi \quad \text{ in } \Omega, \\[1mm] \langle A(u), v - u \rangle + \displaystyle\int_{\Omega} H(x, u, D u)(v - u) \geq 0, & \\[3mm] \forall\; v \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), \quad &v \geq \psi \quad \text{ in } \Omega. \end{cases}
{
u
∈
W
0
1
,
p
(
Ω
)
∩
L
∞
(
Ω
)
,
u
≥
ψ
in
Ω
,
⟨
A
(
u
)
,
v
−
u
⟩
+
∫
Ω
H
(
x
,
u
,
D
u
)
(
v
−
u
)
≥
0
,
∀
v
∈
W
0
1
,
p
(
Ω
)
∩
L
∞
(
Ω
)
,
v
≥
ψ
in
Ω
.
Here,
A
A
is a Leray–Lions type operator, mapping
W_0^{1,p}(\Omega)
W
0
1
,
p
(
Ω
)
into its dual
W^{-1, p'}(\Omega)
W
−
1
,
p
′
(
Ω
)
, while
H(x, u, D u)
H
(
x
,
u
,
D
u
)
grows like
|D u|^p
∣
D
u
∣
p
. The obstacle
\psi
ψ
is a function in
W_0^{1,p}(\Omega) \cap L^\infty(\Omega)
W
0
1
,
p
(
Ω
)
∩
L
∞
(
Ω
)
. Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems