AbstractWe study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$
{
−
Δ
u
=
λ
f
(
v
)
;
Ω
,
−
Δ
v
=
λ
g
(
u
)
;
Ω
,
∂
u
∂
η
+
λ
u
=
0
;
∂
Ω
,
∂
v
∂
η
+
λ
v
=
0
;
∂
Ω
,
where $\lambda >0$
λ
>
0
is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$
R
N
$(N > 1)$
(
N
>
1
)
with smooth boundary ∂Ω, or $\Omega =(0,1)$
Ω
=
(
0
,
1
)
, $\frac{\partial z}{\partial \eta }$
∂
z
∂
η
is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$
f
(
0
)
=
0
=
g
(
0
)
and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$
lim
s
→
∞
f
(
M
g
(
s
)
)
s
=
0
for all $M>0$
M
>
0
. In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.
One Concave-Convex Inequality and Its Consequences
