The existence of at least one positive radial solution of the Neumann problem $$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$−ΔHnu+R(ξ)u=a(|ξ|Hn)|u|p−2u−b(|ξ|Hn)|u|q−2u, is proved on the Heisenberg group $\mathbb{H}^{n}$Hn, via the variational principle, where $a(|\xi |_{\mathbb{H}^{n}})$a(|ξ|Hn), $b(|\xi |_{\mathbb{H}^{n}})$b(|ξ|Hn) are nonnegative radial functions and $R(\xi )$R(ξ) satisfies suitable conditions.
In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized $p(x)$
p
(
x
)
-Laplacian problem $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$
−
Δ
p
(
x
)
u
+
R
(
x
)
u
p
(
x
)
−
2
u
=
a
(
x
)
|
u
|
q
(
x
)
−
2
u
−
b
(
x
)
|
u
|
r
(
x
)
−
2
u
with Dirichlet boundary condition in the unit ball in $\mathbb{R}^{N}$
R
N
(for $N \geq 3$
N
≥
3
), where a, b, R are radial functions.
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