AbstractWe consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$
-
Δ
u
+
g
(
u
)
|
∇
u
|
2
=
λ
f
(
u
)
in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$
Ω
⊂
R
N
($$ N \ge 3$$
N
≥
3
) with zero Dirichlet boundary condition. Here, $$ \lambda $$
λ
is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$
f
:
[
0
,
∞
)
:
(
0
∞
)
is a strictly increasing function of class $$C^1$$
C
1
, and g is a continuous positive decreasing function in $$ (0, \infty ) $$
(
0
,
∞
)
and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$
u
∗
. A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$
h
(
t
)
=
f
(
t
)
e
-
∫
0
t
g
(
s
)
d
s
, nor that the functions $$ gh/h'$$
g
h
/
h
′
or $$ h'' h/h'^2$$
h
′
′
h
/
h
′
2
admit a limit at infinity.
Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients
