Abstract
Abstract
In this paper, we study the p-Laplacian equation with a L
p
-norm constraint:
−
Δ
p
u
=
λ
|
u
|
p
−
2
u
+
μ
|
u
|
q
−
2
u
+
g
(
u
)
in
R
N
,
∫
R
N
|
u
|
p
d
x
=
a
p
,
where N ⩾ 2, a > 0,
1
<
p
<
q
⩽
p
¯
≔
p
+
p
2
N
,
μ
∈
R
,
g
∈
C
(
R
,
R
)
and
λ
∈
R
is a Lagrange multiplier, which appears due to the mass constraint ‖u‖
p
= a. We assume that g is odd and L
p
-supercritical. When
q
<
p
¯
and μ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable μ. When
q
=
p
¯
and μ > 0 or
q
⩽
p
¯
and μ ⩽ 0, with an additional condition of g, we obtain a positive radial ground state if μ lies in a suitable range, by the Schwarz rearrangement and minimax theorems. Via a fountain theorem type argument, with suitable
μ
∈
R
, we obtain infinitely many radial solutions for any N ⩾ 2 and establish the existence of infinitely many nonradial sign-changing solutions for N = 4 or N ⩾ 6. In any case mentioned above, the range of μ depends on the value of a: |μ| can be large if a > 0 is small.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Cited by
4 articles.
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