TITLE:
Existence of Positive Solutions for a Class of Nonlocal Elliptic Differential Equations
AUTHORS:
Jiahua Zhu
KEYWORDS:
Positive Solutions, Bifurcation Methods, Eigenvalues, Kirchhoff-Type Elliptic Equation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.11,
November
11,
2026
ABSTRACT: This paper investigates the existence and multiplicity of positive solutions for the Kirchhoff-type elliptic equation
{
−M(
∫
Ω
|
∇u |
2
dx
)Δu=λf(
x,u
),
x∈Ω,
u(
x
)=0,
x∈∂Ω,
where
λ≥0
is a parameter,
Ω⊂
ℝ
N
(
N≥1
)
is a bounded smooth domain, and
M:[
0,+∞ )→ℝ
and
f:Ω×ℝ→ℝ
are smooth functions satisfying
f(
x,0
)>0
for all
x∈Ω
. By employing the bifurcation method, the existence of positive solutions is successfully proved under appropriate hypotheses regarding
M
and
f
. F Furthermore, when the nonlinearity
f
admits two zeros on the intery
[
0,+∞ )
, further multiplicity results for the positive solutions are derived.