Abstract
In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ)
s
in ℝ
n
, for n ≥ 2, such as
(0.1)
(
−
Δ
)
s
u
+
E
(
x
)
u
+
V
(
x
)
u
q
−
1
=
K
(
x
)
f
(
u
)
+
u
2
s
⋆
−
1
.
$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$
Here, s ∈ (0, 1),
q
∈
2
,
2
s
⋆
$q \in\left[2,2_{s}^{\star}\right)$
with
2
s
⋆
:=
2
n
n
−
2
s
$2_{s}^{\star}:=\frac{2 n}{n-2 s}$
being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝ
n
→ ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ
n
such as
(0.2)
(
−
Δ
)
s
u
+
E
(
x
)
u
+
V
(
x
)
u
q
−
1
=
λ
K
(
x
)
u
r
−
1
,
$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$
where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝ
n
→ ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms
;q,p
(ℝ
n
) as well as their associated compact embedding results.
Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent