Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 21, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

MAXIMUM AND ANTIMAXIMUM PRINCIPLES FOR THE

p-LAPLACIAN WITH WEIGHTED STEKLOV BOUNDARY

CONDITIONS

MABEL CUESTA, LIAMIDI LEADI, PASCALINE NSHIMIRIMANA

Abstract. We study the maximum and antimaximum principles for the p-

Laplacian operator under Steklov boundary conditions with an indefinite weight

−∆pu+ |u|p−2u = 0 in Ω,

|∇u|p−2 ∂u

∂ν
= λm(x)|u|p−2u+ h(x) on ∂Ω,

where Ω is a smooth bounded domain of RN , N > 1. After reviewing some

elementary properties of the principal eigenvalues of the p-Laplacian under
Steklov boundary conditions with an indefinite weight, we investigate the max-

imum and antimaximum principles for this problem. Also we give a character-

ization for the interval of the validity of the uniform antimaximum principle.

1. Introduction

Let Ω be a bounded domain of RN of class C2,α for some 0 < α < 1, N ≥ 1. We
consider the quasilinear problem

−∆pu+ |u|p−2u = 0 in Ω,

|∇u|p−2 ∂u

∂ν
= λm(x)|u|p−2u+ h(x) on ∂Ω.

(1.1)

Here ∆pu := div(|∇u|p−2∇u) is the well known p-Laplacian operator, 1 < p <∞;
m and h are given functions in Cr(∂Ω) for some 0 < r < 1. The weight m can
change sign, and h ≥ 0, h 6≡ 0. We denote by ν = ν(x) the outer normal at x,
defined for all x ∈ ∂Ω and by σ the restriction to ∂Ω of the (N − 1)-Hausdorff
measure, which coincides with the usual Lebesgue surface measure as ∂Ω is regular
enough. All the integral along ∂Ω will be understood with respect to the measure
σ.

Problems of the form (1.1) appears in several branches of pure and applied math-
ematics, such as the theory of quasiregular and quasiconformal mappings in Rie-
mannian manifolds with boundary, non-Newtonian fluids, reaction diffusion prob-
lems, flow through porous media, nonlinear elasticity, glaciology, etc.

The maximum and antimaximum principles for problem (1.1) with m ≡ 1, have
been studied in [3]. The authors proved that every solution of (1.1) is positive if

2010 Mathematics Subject Classification. 35J70.
Key words and phrases. p-Laplacian; Steklov boundary conditions: indefinite weight;

maximum and antimaximum principles.
c©2020 Texas State University.

Submitted November 18, 2019. Published March 2, 2020.

1



2 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

λ ∈ (0, λ1) (maximum principle) and that there exists δ = δ(h) > 0 such that, if
λ ∈ (λ1, λ1 + δ), then every weak solution is negative (antimaximum principle).
Here λ1 denotes the smallest eigenvalue of the eigenvalue problem associated with
(1.1) with m ≡ 1. The authors in [3] also characterize the interval of validity of
the uniform antimaximum principle. A uniform antimaximum principle has also
been proved in [4, 10, 11] for the p-Laplacian operator with Neumann boundary
conditions.

Our main aim in this work is to extend the results proved in [3] to the problem
(1.1) when the weight m is indefinite. Let us define the number

λ̂1(m) := inf
{
‖u‖p1,p;

∫
∂Ω

m|u|p = 1 and u ∈ Q
}

where

Q :=
{
u ∈W 1,p(Ω);∃B(x0, δ) s.t u|B(x0,δ)∩Ω

≡ 0 a.e.
}
.

The norm ‖·‖1,p stand here for the natural norm of W 1,p(Ω). We prove in Theorem

4.4 that the real number λ̂1(m) provides an interval of validity of the uniform
antimaximum principle for (1.1) to the right of λ1(m), where λ1(m) is the first
positive eigenvalue of the eigenvalue problem associated with (1.1). We point out

here that λ̄1(m) ≤ λ̂1(m), where λ̄1(m) is the real number found in [3] for the
validity of the uniform antimaximum principle in the case m ≡ 1, given by

λ̄1(m) := inf
{
‖u‖p1,p;

∫
∂Ω

|u|p = 1 and u vanishes in a ball of Ω
}
.

We will also prove that λ̂1(m) = λ1(m) if 1 < p ≤ N and λ̂1(m) > λ1(m) if

p > N . Furthermore, we prove in Theorem 4.5 that the value λ̂1(m) is the greater
number σ > λ1(m) such that the uniform antimaximun principle holds for any
λ ∈ (λ1(m), σ).

This article is organized as follows. In Section 2, we recall some basic definitions
and we review some properties of the principal eigenvalues of the p-Laplacian under
Steklov boundary conditions with an indefinite weight. We prove in Section 3 some
results concern in maximum principle, existence of solutions and nonexistence of
positive solutions for (1.1). We conclude this paper in Section 4 with some results
on the antimaximum principle and on the uniformity for this principle. Our mean
results of this section are Theorem 4.4 and Theorem 4.5. We finish with some
example in dimension 1.

2. Preliminaries

Throughout this work, m and h are given functions in Cr(∂Ω), for some 0 < r <
1; m± = max{±m(x), 0} and h ≥ 0, h 6≡ 0 a.e.

We denote by W 1,p(Ω) the classical Sobolev space endowed with its natural norm

‖u‖1,p :=
(∫

Ω

(|∇u|p + |u|p)
)1/p

.

The Lebesgue norm of Lp(Ω) will be denoted by ‖ · ‖p, and the one of Lp(∂Ω)
by ‖ · ‖p,∂Ω, for any p ∈ [1,+∞]. If S ⊂ RN is measurable set, |S| denotes the
Lebesgue measure of S and for S ⊂ ∂Ω we will also denote by |S| its σ-mesure.
The weak convergence will be denoted by ⇀ and the strong one by →. Here we



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 3

will denote by p∗ := Np/(N − p)+ the classical critical Sobolev’s exponent, and by
p∗ := (N − 1)p/(N − p)+ the critical Sobolev’s exponent for the trace inclusion.

We are interested in the weak solutions of (1.1), i.e., functions u ∈W 1,p(Ω) such
that ∫

Ω

(|∇u|p−2∇u∇v + |u|p−2uv) = λ

∫
∂Ω

m|u|puv +

∫
∂Ω

hv,

holds for all v ∈W 1,p(Ω).

Remark 2.1. The standard regularity results for quasilinear elliptic pde ensure
that if m,h ∈ Cr(∂Ω) for some 0 < r < 1 then every weak solution of (1.1) lies in
C1,α(Ω). Furthermore, observing that the W 1,p(Ω)−norm of a solution u of (1.1)
can be bounded in terms of ‖u‖∞,∂Ω, ‖m‖∞,∂Ω, ‖h‖∞.∂Ω and |λ| , it follows that if

‖u‖∞,∂Ω, ‖m‖∞,∂Ω, ‖h‖∞.∂Ω, |λ| ≤M

for a constant M > 0, then there exists a constant κ > 0, depending on M,p,Ω,
such that

‖u‖C1,α(Ω) ≤ κ,
see [2, 6] for the details.

Let us summarize some properties of the principal eigenvalues of the eigenvalue
problem associated with problem (1.1),

−∆pu+ |u|p−2u = 0 in Ω,

|∇u|p−2 ∂u

∂ν
= λm(x)|u|p−2u on ∂Ω.

(2.1)

A real number λ is said to be an eigenvalue of (2.1) if and only if there exists
u ∈W 1,p(Ω) \ {0}, called eigenfunction associated with λ, satisfying∫

Ω

(|∇u|p−2∇u∇v + |u|p−2uv) = λ

∫
∂Ω

m|u|p−2uv, (2.2)

for all v ∈ W 1,p(Ω). It is proved in [5] (see also [7] and [12] for a more general
problem) that (2.1) admits two principal eigenvalues which are characterized by

λ1(m) := min{‖u‖p1,p;u ∈W 1,p(Ω), I(u) = 1} > 0; (2.3)

λ−1(m) := −min{‖u‖p1,p;u ∈W 1,p(Ω), I(u) = −1} < 0. (2.4)

where I : W 1.p(Ω)→ R is the C1-functional

I(u) :=

∫
∂Ω

m|u|p. (2.5)

λ1(m) and λ−1(m) are simple, isolated. Moreover since in the case N ≥ 2 there
exists actually two sequences of eigenvalues going one to +∞ and the other to
−∞, we can define the second eigenvalue from the right λ2(m) (resp. the negative
eigenvalue from the left λ−2(m)) of (2.1) as follows:

λ2(m) := min{λ ∈ R;λ eigenvalue and λ > λ1(m)}; (2.6)

λ−2(m) := max{λ ∈ R;λ eigenvalue and λ < λ−1(m)}. (2.7)

See section 5, where we discuss the case N = 1, and let us agree to write λ2(m) =
+∞ (resp. λ−2(m) = −∞ if no eigenvalues greater than λ1(m) (resp. λ−1(m) )
exist. Every eigenfunction u associated with a positive (resp. negative) eigenvalue



4 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

λ 6= λ1(m) (resp. λ 6= λ−1(m)) changes sign. Furthermore, if N is a nodal domain
of u then

|N ∩ ∂Ω| ≥ κ1 > 0, (2.8)

for some constant κ1 > 0 independent of u, see [5]. The following result is a simple
consequence of the characterizations (2.3) and (2.4).

Lemma 2.2. Assume that h ≥ 0, h 6≡ 0. If λ ∈ (λ−1(m), λ1(m)) then there exists
a constant κ > 0 such that

‖u‖p1,p − λI(u) ≥ κ‖u‖p1,p ∀u ∈W 1,p(Ω).

Proof. Assume, by contradiction, that there exists a sequence (un)n∈N∗ ⊂W 1,p(Ω)
with ‖un‖1,p = 1 such that

‖un‖p1,p − λI(un) <
1

n
. (2.9)

Since ‖un‖1,p = 1, then (un)n∈N∗ is bounded in W 1,p(Ω) and there exists a function
u such that un ⇀ u in W 1,p(Ω) and strongly in Lp(∂Ω). Then we obtain

1− λI(u) = lim
n→∞

(1− λI(un)) ≤ 0

and then λI(u) ≥ 1. In particular I(u) 6= 0.
It follows from (2.9) and the weak lower semicontinuity of the norm that

‖u‖p1,p ≤ lim inf
n→∞

‖un‖p1,p ≤ λI(u) (2.10)

Consequently, if I(u) > 0, it follows from (2.3) and (2.10) that

λ ≥
‖u‖p1,p
I(u)

≥ λ1(m),

which is a contradiction. If I(u) < 0, it follows from (2.4) and (2.10) that

λ ≤
‖u‖p1,p
I(u)

≤ λ−1(m),

which is also a contradiction. �

3. Maximum principle and existence of positive solutions

In this section we prove some results on maximum principle for problem (1.1),
existence and uniqueness of solutions, and nonexistence of positive solutions.

Remark 3.1. Let u ∈W 1,p(Ω) be a nonnegative weak solution of (1.1) with h ≥ 0.
Using Harnack’s inequality (see [13, Theorems 5,6 and 9 pages 264-270]) and Hopf
maximum principle (see [15]), its follows that u > 0 a.e. in Ω.

Next, let us recall Picone’s identity, see [1]. Let v > 0 and u ≥ 0 be two
differentiable functions a.e. in Ω and denote

L(u, v) = |∇u|p + (p− 1)
up

vp
|∇v|p − pu

p−1

vp−1
|∇v|p−2∇v · ∇u;

R(u, v) = |∇u|p − |∇v|p−2∇v · ∇
( up

vp−1

)
.

Then Picone’s identity states that

(i) L(u, v) = R(u, v);
(ii) L(u, v) ≥ 0;



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 5

(iii) L(u, v) = 0 in Ω if and only if u = kv for some constant k.

As a consequence of these identities we have the following result.

Lemma 3.2. Let u ∈ C1(Ω) be a weak solution for (1.1) such that u > 0 in Ω.
Then

λI(φ) +

∫
∂Ω

h
|φ|p

up−1
≤ ‖φ‖p1,p (3.1)

for all bounded φ ∈W 1,p(Ω). Moreover the equality holds if and only if φ is scalar
multiplier of u.

Proof. By Picone’s identity we obtain

0 ≤
∫

Ω

L(|φ|, u) =

∫
Ω

R(|φ|, u)

=

∫
Ω

|∇φ|p −
∫

Ω

|∇u|p−2∇u · ∇
( |φ|p
up−1

)
=

∫
Ω

|∇φ|p +

∫
Ω

up−1 |φ|p

up−1
− λ

∫
∂Ω

mup−1 |φ|p

up−1
−
∫
∂Ω

h
|φ|p

up−1

=

∫
Ω

|∇φ|p +

∫
Ω

|φ|p − λ
∫
∂Ω

m|φ|p −
∫
∂Ω

h
|φ|p

up−1

= ‖φ‖p1,p − λI(φ)−
∫
∂Ω

h
|φ|p

up−1
;

(3.2)

and (3.1) holds. Moreover, from assertion (iii) of Picone’s identity we have the
equality in (3.2) if and only |φ| = cu, for some constant c. In particular φ is of
constant sign in Ω. �

The following result states the maximum principle for problem (1.1) for the usual
range of λ.

Theorem 3.3. Assume h ≥ 0, h 6≡ 0. Then the maximum principle for (1.1)
holds if λ ∈ (λ−1(m), λ1(m)), i.e. if u is a weak solution for (1.1) with λ ∈
(λ−1(m), λ1(m)), then u > 0 in Ω.

Proof. Assume by contradiction that u− 6≡ 0 and take v = u− as test function in
(1.1). We have

0 <

∫
Ω

(
|∇u−|p + (u−)p

)
= λ

∫
∂Ω

m(u−)p −
∫
∂Ω

hu−

≤ λ
∫
∂Ω

m(u−)p.

(3.3)

If
∫
∂Ω
m(u−)p > 0, we deduce from the variational characterization of λ1(m) that

(λ1(m)− λ)

∫
∂Ω

m(u−)p ≤ 0

which implies that λ1(m) ≤ λ, we have a contradiction. Similarly, if
∫
∂Ω
m(u−)p <

0, we deduce from the variational characterization of λ−1(m) that

(λ−1(m)− λ)

∫
∂Ω

m(u−)p ≤ 0

which implies that λ−1(m) ≥ λ, and we have a contradiction. Hence, in all cases,
we obtain that u ≥ 0 and the conclusion follows from Remark 3.1. �



6 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

Let us now prove the following uniqueness result. We stress here that the exis-
tence result is well known for any h ∈ Lq(∂Ω) if q > (p∗)

′
. We give here the proof

for the sake of completeness.

Proposition 3.4. Let h ≥ 0, h 6≡ 0 on ∂Ω. Then (1.1) has a unique solution if
λ ∈ (λ−1(m), λ1(m)).

Proof. Let us prove that the energy functional K associated with (1.1)

K(u) =
1

p
‖u‖p1,p −

λ

p
I(u)−

∫
∂Ω

hu.

is coercive and weakly lower semicontinuous. Since λ ∈ (λ−1(m), λ1(m)), it follows
from Lemma 2.2 that

K(u) =
1

p
‖u‖p1,p −

λ

p
I(u)−

∫
∂Ω

hu

≥ κ1

p
‖u‖p1,p − ‖h‖∞‖u‖1,∂Ω

≥ κ

p
‖u‖p1,p − κ1‖u‖1,p →∞ as ‖u‖1,p →∞

where κ1 = c‖h‖∞ with c > 0 the constant from the embedding of W 1,p(Ω) in
L1(∂Ω). We conclude that K is coercive. Now assume that un is a sequence
in W 1,p(Ω) such that un ⇀ u for some u in W 1,p(Ω). Then, from the compact
embedding of W 1,p(Ω) into Lq2(∂Ω), for all q2 ∈ [1, p∗) we can assume that un → u
in Lp(∂Ω) and in L1(∂Ω). Then from the lower semicontinuity of the norm we
obtain

K(u) ≤ lim inf
n→∞

K(un)

and the result follows. Since K is coercive and weakly lower semicontinuous, then
the inf{K(u), u ∈W 1,p(Ω)} is achieved, providing us with a weak solution of (1.1)
(i.e. a critical point of K).

To prove the uniqueness of the solution, assume that u, v ∈ W 1,p(Ω) are two
solutions of (1.1) for a fixed λ ∈ (λ−1(m), λ1(m)). From Theorem 3.3 we have that
u and v are positive and from Lemma 3.2 that

λI(v) +

∫
∂Ω

h
vp

up−1
≤ ‖v‖p1,p = λI(v) +

∫
∂Ω

hv. (3.4)

Hence ∫
∂Ω

h
vp

up−1
≤
∫
∂Ω

hv,∫
∂Ω

hv
(

1− vp−1

up−1

)
≥ 0. (3.5)

Interchanging u and v we also have∫
∂Ω

hu
(

1− up−1

vp−1

)
≥ 0. (3.6)

By adding (3.5) and (3.6), we have∫
∂Ω

h
[
v
(

1− vp−1

up−1

)
+ u
(

1− up−1

vp−1

)]
≥ 0. (3.7)



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 7

Observe that

v
(

1− vp−1

up−1

)
+ u
(

1− up−1

vp−1

)
=
vup−1 − vp

up−1
+
uvp−1 − up

vp−1

=
(up−1 − vp−1)(vp − up)

(uv)p−1
≤ 0.

Thus from (3.7) we obtain∫
∂Ω

h
[
v
(

1− vp−1

up−1

)
+ v
(

1− vp−1

up−1

)]
= 0;

so u = v on the set of positive measure {x ∈ ∂Ω;h(x) 6= 0}. Hence, from (3.4), we
obtain

λI(v) +

∫
∂Ω

h
vp

up−1
= ‖v‖p1,p. (3.8)

Then it follows from the Lemma 3.2 that v = cu for some constant c > 0. Since
u = v on {x ∈ ∂Ω;h(x) 6= 0} then c = 1 and we obtain the desired result. �

Next we prove that there are no positive solutions when the parameter λ lies
outside the interval (λ−1(m), λ1(m)).

Theorem 3.5. Let h ≥ 0, h 6≡ 0 on ∂Ω.

(1) Problem (1.1) has no solution u ≥ 0, u 6≡ 0 if λ 6∈ [λ−1(m), λ1(m)].
(2) Problem (1.1) has no solution if λ = λ1(m) or λ = λ−1(m).

Proof. 1. Assume by contradiction that there exists a nontrivial nonnegative so-
lution u. We deduce from Remark 3.1 that u > 0 in Ω and by Lemma 3.2 one
gets

λI(φ) +

∫
∂Ω

h
|φ|p

up−1
≤ ‖φ‖p1,p (3.9)

for all bounded φ ∈ C1(Ω) and in particular

λI(φ) ≤ ‖φ‖p1,p. (3.10)

Then, by taking any φ ∈ C1(Ω) such that
∫
∂Ω
m|φ|p > 0, it follows from (3.10) and

the variational characterization of λ1(m) that λ ≤ λ1(m). Similarly, by taking any
φ ∈ C1(Ω) satisfying I(φ) < 0 we have that λ ≥ λ−1(m), a contradiction.

2. We only give the proof for the case λ = λ1(m). Assume by contradiction that
there exists a solution u of (1.1) with λ = λ1(m). We claim that u ≥ 0. Indeed, if
not, we take v = u− 6≡ 0 as test function in (1.1) with λ = λ1(m) to obtain

‖u−‖p1,p = λ1(m)I(u−)−
∫
∂Ω

hu−

≤ λ1(m)I(u−)

(3.11)

and from the variational characterization of λ1(m) we have

0 < ‖u−‖p1,p = λ1(m)I(u−). (3.12)

We conclude that the infimum in (2.3) is achieved at u− so u− > 0 in Ω. Besides
from (3.11) we obtain

∫
∂Ω
hu− = 0, a contradiction since h ≥ 0, h 6≡ 0. We have

just proved that u ≥ 0 and hence, by Remark 3.1, u > 0 in Ω. By applying Lemma
3.2 one gets

λ1(m)I(φ) +

∫
∂Ω

h
|φ|p

up−1
≤ ‖φ‖p1,p



8 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

for all φ ∈ C1(Ω).

Finally, by choosing φ = ϕ1 > 0 in Ω, we obtain
∫
∂Ω
h

ϕp1
up−1 ≤ 0, a contradiction.

�

4. Antimaximum principle

In Theorem 4.4 we will prove, for the case p > N , the existence of an interval of
uniformity of this principle for problem (1.1).

The following result shows that the antimaximum principle holds, for a fixed
h ∈ Cr(∂Ω), at the right of λ1(m) (resp. left of λ−1(m)) for λ sufficiently close to
λ1(m) (resp. λ−1(m)) and for any p > 1.

Theorem 4.1. Let h ≥ 0, h 6≡ 0. Then there exists δ = δ(h) > 0 such that if
λ ∈ (λ1(m), λ1(m) + δ) then any solution u of (1.1) satisfies u < 0 in Ω.

Similarly, there exists δ′ = δ′(h) > 0 such that if λ ∈ (λ−1(m) − δ′, λ−1(m)),
any solution u of (1.1) satisfies u < 0 in Ω.

Proof. We only give the proof for the case λ ∈ (λ1(m), λ1(m) + δ). We assume by
contradiction that there exists a sequence (λk, uk) ∈ R×W 1,p(Ω) with λk > λ1(m),
λk → λ1(m), uk a solution of (Pλk,h) and such that

uk(xk) ≥ 0 for some xk ∈ Ω.

Two alternatives can arise:
(a) ‖uk‖∞,∂Ω ≤ κ2, with κ2 some positive constant. It follows that uk is also

bounded in L∞(Ω) and in C1,α(Ω), see Remark 2.1. Then, using the compact
embedding of C1,α(Ω) into C1(Ω) we obtain, up to a subsequence, that uk → u for
some function u in C1(Ω). Passing to the limit in (Pλk,h) we obtain that u is a
weak solution of (1.1) for λ = λ1(m), a contradiction with Theorem 3.5 (2).

(b) ‖uk‖∞,∂Ω → ∞. Setting wk := uk
‖uk‖∞,∂Ω

, then ‖wk‖∞,∂Ω = 1 and it follows

(using Remark 2.1 with hk := h

‖uk‖p−1
∞,∂Ω

) that wk lies in C1,α(Ω). Moreover there

exists a constant C > 0 such that ‖wk‖C1,α(Ω) ≤ C. Thus, there exists a function

w such that, for a subsequence, wn → w in C1(Ω). In particular w 6≡ 0 since
‖w‖∞,∂Ω = 1. Hence, passing to the limit, we obtain that w is an eigenfunction

associated with the eigenvalue λ1(m) of (2.1). Consequently w > 0 or w < 0 in Ω.
If w > 0, then for k large enough we have uk > 0 and this contradicts Theorem
3.5(1). If w < 0, then for k large enough we have uk < 0 which contradicts the
existence of xk. �

Notice that, a priori, the value δ of Theorem 4.1 depends of the function h. If this
is not so, we say that the antimaximum principle is uniform on (λ1(m), λ1(m) + δ).
In the following, we study the validity of the uniform antimaximum principle and
we will give a variational characterization of the greatest value δ for which the
uniform antimaximum principle holds in (λ1(m), λ1(m) + δ) if p > N .

Following [3, 4, 10] we introduce the values λ̂1(m) and λ̂−1(m):

λ̂1(m) := inf
{
‖u‖p1,p; I(u) = 1 and u ∈ Q

}
(4.1)

λ̂−1(m) := − inf
{
‖u‖p1,p; I(u) = −1 and u ∈ Q

}
, (4.2)

where

Q :=
{
u ∈W 1,p(Ω);∃B(x0, r) s.t u|B(x0,r)∩Ω

≡ 0 a.e.
}

with x0 ∈ ∂Ω



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 9

Clearly λ1(m) ≤ λ̂1(m) (resp. λ−1(m) ≥ λ̂−1(m)). In the following two lemmae

we discuss whenever λ1(m) is different or equal to λ̂1(m).

Lemma 4.2. Assume 1 < p ≤ N and let λ̂1(m), λ̂−1(m) be defined in (4.1) et

(4.2). Then λ1(m) = λ̂1(m) and λ−1(m) = λ̂−1(m).

Proof. We distinguish two cases:
Case (i) p < N . We define, as in [3] or [4], the sequence of functions yk defined

for all x ∈ RN by

yk(x) :=


1 if |x| ≥ 1

k ,

2k|x| − 1 if 1
2k < |x| <

1
k ,

0 if |x| ≤ 1
2k

It is not difficult to prove that yk → 1 as k →∞ in W 1,p
loc (RN ) using∫

RN

∣∣∂yk
∂xi

∣∣p =

∫
1
2k<|x|<

1
k

∣∣∂yk
∂xi

∣∣p ≤ Ckp−N → 0 as k →∞. (4.3)

for some constant C.
On the other hand, let us assume without loss of generality that 0 ∈ ∂Ω and

let ϕ1 be the positive eigenfunction associated with λ1(m) satisfying I(ϕ1) = 1.
Then the sequence zk := ϕ1(x)yk(x) vanishes in the set B

(
0, 1

2k

)
∩ ∂Ω and clearly

zk ∈ Lp(Ω). Moreover, since ϕ1 lies in C1,α(Ω) and ‖ϕ1‖C1,α(Ω) ≤ C, for some

constant C > 0, we infer that∫
Ω

∣∣∂zk
∂xi

∣∣p ≤ 2p
∫

Ω

∣∣∂ϕ1

∂xi
yk
∣∣p + 2p‖ϕ1‖p∞

∫
Ω

∣∣∂yk
∂xi

∣∣p,
and therefore ∂zk

∂xi
∈ Lp(Ω) and zk ∈W 1,p(Ω). On the other hand, we have

|zk − ϕ1|p = |ϕ1yk − ϕ1|p ≤ ϕp1 ∈ L1(Ω);

|zk − ϕ1|p = |ϕ1yk − ϕ1|p
k→∞−−−−→ 0 a.e.;∣∣∣∂ϕ1

∂xi
yk −

∂ϕ1

∂xi

∣∣∣p ≤ ∣∣∣∂ϕ1

∂xi

∣∣∣p ∈ L1(Ω);∣∣∣∂ϕ1

∂xi
yk −

∂ϕ1

∂xi

∣∣∣p → 0 as k →∞;

and by Lebesgue’s Dominated Convergence Theorem it follows that zk → ϕ1 in
W 1,p(Ω) as k →∞. Hence

I(zk)→ I(ϕ1) = 1 as k →∞

and in particular, for k large enough one has I(zk) > 0. Then, from the definition

(4.1) of λ̂1(m), we have

λ̂1(m) ≤
‖zk‖p1,p
I(zk)

→ ‖ϕ1‖p1,p = λ1(m).

Case (ii) p = N . In this case we define instead

yk(x) :=


1− 2

k if |x| ≥ 1
k ,

|x|δk − 1
k if

(
1
k

)1/δk < |x| < 1
k ,

0 if |x| ≤
(

1
k

)1/δk ,



10 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

where δk satisfies ( 1
k )δk = 1 − 1

k ; (δk = 1 − ln(k−1)
ln(k) → 0 as k → ∞). It is easy to

show that yk → 1 as k →∞ a.e. in W 1,p
loc (Ω) since∫

RN

∣∣∂yk
∂xi

∣∣p =

∫
( 1
k )1/δk<|x|< 1

k

∣∣∣δk|x|(δk−1) xi
|x|

∣∣∣p
≤ Cδpk

∫
( 1
k )

1/δk<r< 1
k

r(δk−1)prN−1dr

=
Cδpk

(δk − 1)p+N

[(1

k

)δkp − (1

k

)p]
which converges to 0 as k → ∞. The sequence zk := ϕ1yk vanishes in the set
B
(
0, ( 1

k )1/δk
)
∩ ∂Ω and clearly zk lies in Lp(Ω). Moreover, using the arguments of

(i), we have zk → ϕ1(x) as k →∞ a.e. in W 1,p(Ω). Hence, from the definition (4.1)

of λ̂1(m), one deduces that

λ̂1(m) ≤
‖zk‖p1,p
I(zk)

→ ‖ϕ1‖p1,p = λ1(m);

as k →∞, and we obtain the conclusion. �

Proposition 4.3. Let p > N .
(a) It holds

λ̂1(m) = inf{‖u‖p1,p; I(u) = 1 and u ∈ Q0}, (4.4)

λ̂−1(m) = − inf{‖u‖p1,p; I(u) = −1 and u ∈ Q0} (4.5)

where

Q0 := {u ∈W 1,p(Ω);∃x0 ∈ ∂Ω s.t. u(x0) = 0}.
(b) The infima in (4.4) and (4.5) are achieved.

(c) λ1(m) < λ̂1(m) and λ̂−1(m) < λ−1(m).
(d) If û is a minimiser in (4.4) (resp. (4.5)), then û vanishes exactly at one

point on ∂Ω and û does not change sign on ∂Ω.

(e) λ̂1(m) < λ2(m) where λ2(m) is the eigenvalue defined in (2.6). Respectively,

λ̂−1(m) > λ−2(m), where λ−2(m) is the eigenvalue defined in (2.7).

Proof. We only give the proofs that concern λ̂1(m).
(a) Let φ ∈ C1(Ω) satisfies I(φ) > 0 and assume that φ(x0) = 0 for some

x0 ∈ ∂Ω. For any fixed ε > 0 let us define

φε := max{|φ|, ε} − ε.

Clearly, φε → |φ| in W 1,p(Ω) as ε→ 0. By continuity, there exists r > 0 such that
|φ(x)| < ε for all x ∈ B(x0, r)∩Ω, and therefore φε(x) = 0 for all x ∈ B(x0, r)∩Ω.

(b) The proof is standard and uses the compact embedding of W 1,p(Ω) in C(Ω)
to assure that a weak limit of any minimizing sequence must vanish somewhere on
∂Ω.

(c) Assume that λ1(m) = λ̂1(m). From (b), λ̂1(m) is achieved at some u0 and
consequently u0 is an eigenfunction of (2.1) associated with λ1(m). But this is

impossible since u0 is vanishes somewhere in ∂Ω. Hence λ1(m) < λ̂1(m).
(d) Let us now prove that the minimiser vanishes exactly at one point on ∂Ω.

Set û the minimiser of λ̂1(m) and assume that û(x0) = 0 for some x0 ∈ ∂Ω. We



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 11

can assume that û ≥ 0 by changing û by |û| if needed. Then the definition (4.4) is
equivalent to the following

λ̂1(m) := inf
u∈A
‖u‖p1,p (4.6)

where

A := {u ∈W 1,p(Ω); I(u) = 1 and u(x0) = 0}.
Assume by contradiction that there exists x1 6= x0 ∈ ∂Ω with û(x1) = 0 and set

B := {u ∈W 1,p(Ω); I(u) = 1 and u(x1) = 0}

so we also have

λ̂1(m) = inf
u∈B
‖u‖p1,p.

Let us now denote by

Ψ(u) := ‖u‖p1,p; ψ1(u) := I(u)− 1, ψ2(u) := u(x1).

By Lagrange’s Multipliers Theorem there exists (β1, β2) ∈ R2 such that

Ψ ′(w)(v) = β1ψ
′
1(w)(v) + β2ψ

′
2(w)(v)

= β1ψ
′
1(w)(v) + β2v(x1) ∀v ∈W 1,p(Ω).

(4.7)

Taking v = w in (4.7) we obtain that β1 = λ̂1(m). Similarly there exists γ2 ∈ R
such that

Ψ ′(w)(v) = λ̂1(m)ψ′1(w)(v) + γ2v(x0) ∀v ∈W 1,p(Ω). (4.8)

and therefore

β2v(x1) = γ2v(x0), ∀v ∈W 1,p(Ω) (4.9)

Taking v ≡ 1 in (4.9) one sets β2 = γ2 and since (4.9) holds for all v ∈ W 1,p(Ω),

we deduce that β2 = γ2 = 0. Consequently it comes from (4.7) that λ̂1(m) is a
principal eigenvalue of (1.1) and w is a nonnegative eigenfunction associated with

λ̂1(m). By Remark 3.1, w > 0 in Ω, a contradiction.
We have just prove that w, and therefore û, vanishes only once on ∂Ω.
Now, let us show that û does not change sign on ∂Ω. Assume that û+ 6≡ 0, û− 6≡ 0

and say û(x1) = 0 for some x1 ∈ ∂Ω. Then taking v = û+ in (4.9), one gets that

0 < ‖û+‖p1,p = λ̂1(m)I(û+),

so the function û+

I(û+)1/p is a minimizer in (4.4). Hence û+ vanishes only at x1 which

implies û ≥ 0 on ∂Ω.
(e) Let ϕ2 be an eigenfunction associated with λ2(m). By (2.8) we know that

ϕ2 vanishes somewhere on ∂Ω. Thus ϕ2 is an admissible function in the definition

(4.4) of λ̂1(m) and then

λ̂1(m) ≤
‖ϕ2‖p1,p
I(ϕ2)

= λ2(m).

If λ̂1(m) = λ2(m) then ϕ2 would be a minimiser in (4.4) and therefore it must have
a constant sign on ∂Ω, according to (c), a contradiction. �

With the previous results in hand, we can give an interval where the uniform
antimaximum principle holds.



12 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

Theorem 4.4. Let p > N and let h ≥ 0, h 6≡ 0. If u is a solution of (1.1)

with λ ∈
(
λ1(m), λ̂1(m)

]
then u < 0 in Ω. Similarly any solution u of (1.1) with

λ ∈
[
λ̂−1(m), λ−1(m)

)
is negative in Ω.

Proof. Let u be a solution of (1.1) with λ ∈
(
λ1(m), λ̂1(m)

]
, then u− 6≡ 0 in Ω by

Theorem 3.5. Let us take v = u− as test function in (1.1) to get

0 < ‖u−‖p1,p = λI(u−)−
∫
∂Ω

hu− ≤ λI(u−). (4.10)

In particular I(u−) > 0. Let us first show that u < 0 on ∂Ω. Indeed, if λ < λ̂1(m),
we have from (4.10)

‖u−‖p1,p
I(u−)

≤ λ < λ̂1(m) = inf
v∈A
‖v‖p1,p, ∀x0 ∈ ∂Ω.

So u− 6∈ A and we conclude that u− does not vanish anywhere on ∂Ω, that is, u < 0

on ∂Ω. If λ = λ̂1(m) and we assume by contradiction that u− vanish somewhere on

∂Ω, hence, from the one hand u− is a minimizer for λ̂1(m) according to Proposition
4.3(a) and from the other hand, using (4.10) we have

0 = ‖u−‖p1,p − λ̂1(m)I(u−) = −
∫
∂Ω

hu−.

We deduce from this relation that u− vanishes on the set of positive measure {x ∈
∂Ω;h(x) > 0} which is a contradiction with Proposition 4.3(c) (minimizers of λ̂1(m)
vanish only once).

Next we prove that u < 0 in Ω. Since u < 0 on ∂Ω one has that u+ ∈ W 1,p
0 (Ω).

Take then v := u+ in the weak form of (1.1) to obtain

‖u+‖p1,p = λI(u+) +

∫
∂Ω

hu+ = 0. (4.11)

Consequently u+ ≡ 0 in Ω and so u ≤ 0 in Ω. Using the well know Harnack’s
inequality [13, Theorem 5] we deduce that u < 0 in Ω and then u < 0 in Ω. �

Finally we prove that the value λ̂1(m) (resp. λ̂−1(m).) is optimal in the sense

that the antimaximum principle holds to the right of λ̂1(m) and that the uniform

antimaximum principle fails to the right of λ̂1(m) + δ for any δ > 0.

Theorem 4.5.

(1) For any h ≥ 0, h 6≡ 0 there exists δ = δ(h) > 0 such that if λ ∈(
λ̂1(m), λ̂1(m) + δ

)
, every solution u of (1.1) satisfies u < 0 in Ω.

(2) Given δ > 0, there exists h ∈ Cr(∂Ω) satisfying h ≥ 0, h 6≡ 0 such that for

all λ > λ̂1(m) + δ problem (1.1) does not admit a negative solution.
In particular for all δ > 0, the uniform antimaximum principle does not

hold in (λ̂1(m), λ̂1(m) + δ).

Similar results can be stated to the left of λ̂−1(m).

Proof. (1) We assume here that p > N as in the case 1 < p ≤ N , λ̂1(m) = λ1(m)
and the result is proved in Theorem 4.1. The proof follows the same pattern of the
one in the proof of Theorem 4.1 and we just indicate the changes needed in the
contradiction argument. In alternative (a), passing to the limit in (Pλk,h) one gets



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 13

that u is a weak solution of (1.1) for λ = λ̂1(m). It follows from Theorem 4.4 that
u < 0 in Ω and consequently uk < 0 in Ω for k large enough (since the convergence
is in C1(Ω)), a contradiction with the existence of xk. In alternative (b), passing

to the limit we obtain that w is an eigenfunction associated with λ̂1(m). Since

‖w‖∞,∂Ω = 1 then w 6≡ 0 and therefore λ̂1(m) is an eigenvalue of (1.1) and w an

eigenfunction associated with λ̂1(m), a contradiction with Proposition 4.3 (e).
(2) Let δ > 0 be fixed and assume by contradiction that for any h ≥ 0, h 6≡ 0,

there exists λ(h) > λ̂1(m) + δ such that (Pλ(h),h) admits a solution uh ∈ C1(Ω)

such that uh < 0 in Ω. Let φ ∈ W 1,p(Ω) satisfies I(φ) > 0 and assume that there
exists x0 ∈ ∂Ω and there exists r > 0 such that φ(x) = 0 a.e. in B(x0, r) ∩ Ω.
Choose h ≥ 0, h 6≡ 0 satisfying

supp∂Ω h ⊂ B(x0, r) ∩ ∂Ω. (4.12)

By applying Lemma 3.2 to v = −uh > 0 (which is a solution of problem (1.1) with
λ = λ(h) and −h instead of h, we obtain

(λ̂1(m) + δ)I(φ) < λ(h)I(φ) ≤ ‖φ‖p1,p
which implies

λ̂1(m) + δ ≤
‖φ‖p1,p
I(φ)

,

and taking the infimum over all φ ∈W 1,p(Ω) satisfying I(φ) > 0 and vanishing on
B(x0, r) ∩ Ω, for some x0 ∈ ∂Ω, we obtain

λ̂1(m) + δ ≤ λ̂1(m),

which is a contradiction. �

5. Spectra in dimension 1

5.1. Case p = 2. A simple computation shows that in the case N = 1, p = 2, there
are only two eigenvalues for the Steklov problem. Take for instance Ω = (0, 1) and

m(x) =

{
−1 if x = 0;

1 if x = 1.

Hence the only eigenvalues of the eigenvalue problem

−u′′ + u = 0 in (0, 1);

−u′(0) = λm(0)u(0),

u′(1) = λm(1)u(1),

are λ−1(m) = −1 and λ1(m) = 1. Let

α = inf{‖u‖21,2;u ∈ H1 and u(1) = 1}, β = inf{‖u‖21,2;u ∈ H2 and u(0) = 1},

where

H1 = {u ∈ H1((0, 1)); I(u) = 1 and u(0) = 0},
H2 = {u ∈ H1((0, 1)); I(u) = 1 and u(1) = 0}.

Then

λ̂1(m) = inf{‖u‖21,2;u ∈ H1((0, 1)), I(u) = 1, u(0) = 0 or u(1) = 0} = min{α, β}



14 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

By a simple computation we obtain

α =
e+ e−1

e− e−1
= β = λ̂1(m).

Furthermore, if h > 0 is a function defined on the boundary of Ω = (0, 1) by

h(x) =

{
a if x = 0;

b if x = 1,

then it results that, if λ > 1, the (unique) solution u of

−u′′ + u = 0 in (0, 1);

−u′(0) = λm(0)u(0) + a,

u′(1) = λm(1)u(1) + b.

is non-positive if and only if

1 < λ <
2b

a(e− e−1)
+
e+ e−1

e− e−1
.

Then there is an uniform antimaximum principle for λ ∈
(
1, e+e

−1

e−e−1

]
.

5.2. General case. Let us consider (1.1) in dimension 1 for the weight m ≡ 1, i.e.

(|u′|p−2u′)′|u|p−2u in Ω = (0, 1)

|u′(0)|p−2u′(0) = −λ|u(0)|p−2u(0)

|u′(1)|p−2u′(1) = λ|u(1)|p−2u(1)

(5.1)

First we look for positive solution u of (5.1) such that

u(0) = u(1), u′(
1

2
) = 0.

From (5.1) we obtain

− |u
′(t)|p

p′
+
|u(t)|p

p
= C, ∀t ∈ (0, 1) (5.2)

where p′ = p
p−1 and the constant C is such that

C = −|u
′(0)|p

p′
+
|u(0)|p

p
= (u(0))p

[1
p
− λ

p
p−1

p′
]

= −|u
′(1/2)|p

p′
+
|u(1/2)|p

p

(5.3)

Let us assume that u(1/2) = 1. Then C = 1
p and

u(0) =
(

1− (p− 1)λ
p
p−1

)−1/p

. (5.4)

Moreover, using the fact that u′(t) < 0 for all t ∈ (0, 1
2 ), from (5.2) we obtain

− du

(|u|p − 1)
1/p

= (p− 1)−1/pdt (5.5)

Hence ∫ u(0)

1

dz

(|z|p − 1)1/p
=

1

2
(p− 1)−1/p



EJDE-2020/21 MAXIMUM AND ANTIMAXIMUM PRINCIPLES 15

or equivalently

u(0) = Λp[
1

2
(p− 1)−1/p] (5.6)

where we denote by Λp : R→ R the function defined implicitly by

Λp(t) = y ⇐⇒ t =

∫ y

1

dz

(|z|p − 1)1/p
(5.7)

From (5.4) and (5.6) we obtain

λ1 = λ =
[ 1

p− 1

(
1−

(
Λp
[1
2

(p− 1)−1/p
])−p)] p−1

p

(5.8)

On another hand, since u′(t) > 0 for all t ∈ ( 1
2 , 1), we deduce from (5.2) that

u(s) = ϕ1(s) = Λp
[
(p− 1)−1/p|s− 1

2
|
]
, ∀s ∈ (0, 1)

Now we look for a solution u = ϕ2 of (5.1) which changes sign on (0, 1) such that
u(1/2) = 0 and u′(1/2) = 1. From (5.2) we deduce that

−|u
′(t)|p

p′
+
|u(t)|p

p
= − 1

p′
= −|u

′(0)|p

p′
+
|u(0)|p

p

= |u(0)|p[ 1
p
− λ

p
p−1

p′
]

(5.9)

and consequently, since u′(t) > 0 for all t ∈ (0, 1), we have

u′(t) =
(
1 +
|u(t)|p

p− 1

)1/p
.

Hence
1

2
=

∫ 0

u(0)

dz

(1 + |z|p
p−1 )1/p

= (p− 1)1/p

∫ −u(0)(p−1)−1/p

0

dz

(1 + |z|p)1/p

(5.10)

Similarly we define Φp(t) = y implicitly by

Φp(t) = y ⇐⇒ t =

∫ y

0

dv

(1 + |v|p)1/p
= (p− 1)−1/p

∫ (p−1)1/py

0

dv

(1 + |v|p
p−1 )1/p

.

Hence from (5.9) and (5.10) we deduce that[
(p− 1)λ

p
p−1 − 1

]−1/p
= −u(0)(p− 1)−1/p = Φp

[1
2

(p− 1)−1/p
]
,

so

λ2 = λ =
{ 1

p− 1

[
1 +

(
Φp[

1

2
(p− 1)−1/p]

)−p]} p−1
p

,

ϕ2(s) = u(s) = (p− 1)1/PΦp[(p− 1)1/P (s− 1

2
)], ∀s ∈ (0, 1).

It remains to explain the value of λ̂1(m). Since

λ̂1(m) = inf{‖u‖p1,p;u(0) = 0 and u(1) = 1} = ‖û‖p1,p



16 M. CUESTA, L. LEADI, P. NSHIMIRIMANA EJDE-2020/21

it follows that û is solution of the problem

(|û′|p−2û′)′ = |û|p−2û

û(0) = 0, û(1) = 1
(5.11)

Hence

− |û
′|p

p′
+
|û|p

p
= C = −|û

′(0)|p

p′
=

1

p
− |û

′(1)|p

p′
(5.12)

From (5.12) we obtain

1 =

∫ 1

0

du

( |u|
p

p−1 − Cp′)1/p
= (p− 1)1/p

∫ (−Cp)−1/p

0

dt

(|t|p + 1)1/p
,

which is equivalent to

Φp[(p− 1)−1/p] = (−Cp)−1/p (5.13)

Multiplying (5.11) by û, integrating by parts and using (5.12) we have

λ̂1(m) = ‖û‖p1,p
= (û′(1))

p−1

= (p− 1)−(p−1)/p (1− Cp)(p−1)/p

Finally (5.13) leads to

Φp[(p− 1)−1/p] =
[
− 1 + (p− 1)

(
λ̂1(m)

) p
p−1
]−1/p

and hence

λ̂1(m) =
{ 1

p− 1

[
1 +

(
Φp[(p− 1)−1/p]

)−p]} p−1
p

Some properties of Φp and Λp can be found in [8, 9, 14]

Acknowledgements. P. Nshimirimana was supported by the Centre d’Excellence
Africain en Sciences Mathématiques et Applications (CEA-SMA) through IMSP.
This work was partially carried out while the she was visiting the LMPA of the Uni-
versité du Littoral Côte d’Opale (ULCO). She would like to express her gratitude
to these institutions.

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Mabel Cuesta

Université du Littoral ULCO, LMPA, 50 rue F. Buisson 62220 Calais, France

Email address: mabel.cuesta@univ-littoral.fr

Liamidi Leadi

Université d’Abomey Calavi, FAST, IMSP, Porto-Novo, Bénin
Email address: leadiare@imsp-uac.org

Pascaline Nshimirimana
Université d’Abomey Calavi, FAST, IMSP, Porto-Novo, Bénin

Email address: pascaline.nshimirimana@imsp-uac.org


	1. Introduction
	2. Preliminaries
	3. Maximum principle and existence of positive solutions
	4. Antimaximum principle
	5. Spectra in dimension 1
	5.1. Case p=2
	5.2. General case
	Acknowledgements

	References