This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).
We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.
Abstract
In this study, we consider the following quasilinear Choquard equation with singularity
−
Δ
u
+
V
(
x
)
u
−
u
Δ
u
2
+
λ
(
I
α
∗
∣
u
∣
p
)
∣
u
∣
p
−
2
u
=
K
(
x
)
u
−
γ
,
x
∈
R
N
,
u
>
0
,
x
∈
R
N
,
\left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right.
where
I
α
{I}_{\alpha }
is a Riesz potential,
0
<
α
<
N
0\lt \alpha \lt N
, and
N
+
α
N
<
p
<
N
+
α
N
−
2
\displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2}
, with
λ
>
0
\lambda \gt 0
. Under suitable assumption on
V
V
and
K
K
, we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as
λ
→
0
\lambda \to 0
.
Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems