Glasgow Math. J. 51 (2009) 513–524. C 2009 Glasgow Mathematical Journal Trust.
doi:10.1017/S001708950900514X. Printed in the United Kingdom

A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS
OF p-LAPLACIAN TYPE WITH SIGN-CHANGING
NONLINEARITIES
NGUYEN THANH CHUNG
Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet,
Dong Hoi, Quang Binh, Vietnam
e-mail:

´ˆ C ANH NGO
ˆ
and QUO
Department of Mathematics, College of Science, Vietnam National University, Hanoi, Vietnam
Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543
e-mail: bookworm
(Received 11 July 2008; accepted 22 December 2008)

Abstract. Using variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the form
−div(a(x, ∇u)) = λf (x, u) in

,

where
is a bounded domain in ‫ޒ‬N , N 3, f is a sign-changing Carath´eodory
function on × [0, +∞) and λ is a positive parameter.
2002 Mathematics Subject Classification. 35J20, 35J60, 35J65, 58E05.
1. Introduction. This paper deals with the non-existence and multiplicity of
non-negative, non-trivial solutions to the following problem,
−div(a(x, ∇u)) = λf (x, u) in

The authors obtained the existence of a weak solution by using a variant of the


´ˆ C ANH NGO
ˆ
NGUYEN THANH CHUNG AND QUO

514

Mountain pass theorem introduced in [3]. Then, H. Q. Toan and Q.-A. Ngoˆ [12] gave
some multiplicity results in the case when f (x, u) = h(x)|u|r−1 u + g(x)|u|s−1 u. Using
the Mountain pass theorem in [3] combined with Ekeland’s variational principle in [5]
they proved that problem (1.3)–(1.4) has at least two weak solutions.
Motivated by K. Perera [10] and M. Mih˘ailescu and V. R˘adulescu [7], the goal of
this work is to investigate the problem (1.1)–(1.2) with positive parameter λ and the
sign-changing nonlinearity f . We also do not require that the nonlinear term f verifies
the Ambrosetti–Rabinowitz condition as in [4, 12].
In order to state our main result, let us introduce the following hypotheses on
problem (1.1)–(1.2).
Assume that N 3 and 2 p < N. Let
be a bounded domain with a
smooth boundary ∂ . Consider that a : × ‫ޒ‬N → ‫ޒ‬N , a = a(x, ξ ), is the continuous
derivative with respect to ξ of the continuous function A : × ‫ޒ‬N → ‫ޒ‬, A = A(x, ξ ),
that is, a(x, ξ ) = ∂A(x, ξ )/∂ξ and A(x, 0) = 0 for a.e. x ∈ . Assume that there are
positive constant c0 and two non-negative measurable functions h0 , h1 such that
h0 ∈ Lp/p−1 ( ), h1 ∈ L1loc ( ), h1 (x) 1, a.e. x ∈ . Suppose that a and A satisfy the
following hypotheses.
(A1 ) |a(x, ξ )| c0 (h0 (x) + h1 (x)|ξ |p−1 ) for all ξ ∈ ‫ޒ‬N , a.e. x ∈
(A2 ) The following inequality holds


EXAMPLE 1.
1. Let
A(x, ξ ) =

h(x) p
|ξ | ,
p

a(x, ξ ) = h(x)|ξ |p−2 ξ,

with p 2 and h ∈ L1loc ( ). Then we get the operator div(h(x)|∇u|p−2 ∇u), and
if h(x) ≡ 1 in we conclude the well-known p-Laplacian operator
pu

:= div(|∇u|p−2 ∇u)

as in [8, 10].
2. Let
A(x, ξ ) =

p
h(x)
(1 + |ξ |2 ) 2 − 1
p


MULTIPLICITY RESULT FOR A CLASS OF p-LAPLACIAN-TYPE EQUATIONS

with p


0 uniformly in x, where F(x, t) = 0 f (x, s) ds.
(F3 ) lim sup F(x,t)
tp
t→∞

Let W 1,p ( ) be the usual Sobolev space and W0 ( ) be the closure of C0∞ ( )
under the norm
1,p

|∇u| dx

u =

p

1
p

.

1,p

We now consider the following subspace of W0 ( ):
1,p

H := u ∈ W0 ( ) :

h1 (x)|∇u|p dx < +∞ .

Then H is an infinite dimensional Banach space with respect to the norm (see [4])

A(x, ∇u) dx ,

I(u) = λ

F(x, u) dx ,

u ∈ H.

(1.6)
1,p

Since h0 ∈ Lp/p−1 ( ), then the value λ (u) may be infinity for some u ∈ W0 ( ), that
1,p
is, the functional may not be defined throughout W0 ( ). In order to overcome this
1,p
difficulty, we choose the subspace H of W0 ( ).


516

´ˆ C ANH NGO
ˆ
NGUYEN THANH CHUNG AND QUO

DEFINITION 1. We say that u ∈ H is a weak solution of problem (1.1)–(1.2) if and
only if
a(x, ∇u)∇ϕ dx −λ

f (x, u)ϕ dx = 0


i

p∗

hold true.
h1 (x)|∇u|p dx < +∞ for any u ∈ C0∞ ( ) and h1 ∈ L1loc ( ), we have
(iii) Since
∞
C0 ( ) ⊂ H.
The main result for the existence of solutions of (1.1) can be formulated as follows.
THEOREM 1. Under hypotheses (A1 )–(A4 ) and (F1 ), there exists a positive constant
λ such that for all λ ∈ (0, λ), problem (1.1)–(1.2) has no weak solution.
THEOREM 2. Under hypotheses (A1 )–(A4 ) and (F1 )–(F3 ), there exists a positive
constant λ such that for all λ λ, problem (1.1)–(1.2) has at least two distinct nonnegative, non-trivial weak solutions.
To prove Theorem 2, we first prove that the functional associated to the problem
(1.1)–(1.2) is bounded from below and coercive, and thus the first weak solution is
obtained due to a variant of the minimum principle which we will prove in the next
section (see Theorem 4). To obtain the second solution to the problem (1.1)–(1.2), we
shall use a variant of the mountain pass theorem due to Duc (see Proposition 1).
may not be
2. Auxiliary results. Due to the presence of h1 , functional
continuously Fr´echet differentiable functionals on H. This means that we cannot apply
the classical Mountain pass theorem by Ambrosetti–Rabinowitz (see [1] for details).
To overcome this difficulty, we shall use a weak version of the Mountain pass theorem
introduced by Duc [3]. Now we introduce the following concept of weakly continuously
differentiability due to Duc.
DEFINITION 2. Let F be a map from a Banach space X to ‫ޒ‬. We say that F is
weakly continuously differentiable on X if and only if the following two conditions are
satisfied:
(i) For any u ∈ X there exists a linear map DF(u) from X to ‫ ޒ‬such that

DF(un ) → 0 in X

possesses a convergent subsequence. If this is true at every level c then we simply say
that F satisfies the Palais-Smale condition (denoted by (PS)).
DEFINITION 5. We say that F satisfies the Cerami condition at level c ∈ ‫( ޒ‬denoted
by (C)c ) if any sequence {un } ⊂ X for which
F(un ) → c

and

(1 + xn )DF (un ) → 0 in X

possesses a convergent subsequence. If this is true at every level c, then we simply say
that F satisfies the Cerami condition (denoted by (C)).
In the proof of our main theorems, we shall use the following results which is
proved in [9]. We will recall its proof for completeness.
THEOREM 3 (see [9]). Let F ∈ Cw1 (X) where X is a Banach space. Assume that
(i) F is bounded from below, c = inf F,
(ii) F satisfies the (PS) condition.
Then c is a critical value of F (i.e. there exists a critical point u0 ∈ X such that F(u0 ) = c)
Proof of Theorem 3. Let c be an arbitrary real number. Before proving the theorem,
we need the following notation:
F c = {u ∈ X|F(u)

c}.

Let us assume, by negation, that c is not a critical value of F. Then, Theorem 2.2 in [13]
implies the existence of ε > 0 and η ∈ C([0, +∞) × X, X) satisfying η(1, F c+ε ) ⊂ F c−ε .
This is a contradiction since F c−ε = ∅ due to the fact that c = inf F.
REMARK 3. By Corollary 2.1.1 in [6], if F : X → ‫ ޒ‬is a locally Lipschitz, bounded

Assume that the set
G = {ϕ ∈ C([0, 1], X) : ϕ(0) = 0, ϕ(1) = z0 }
is not empty. Put
β := inf {max F(ϕ([0, 1])) : ϕ ∈ G} .
Then β

α and β is a critical value of F.

For the use of Proposition 1, we refer the reader to [3, 12, 13]. We end
this section by studying some certain properties of the functional λ given
by (1.5) but we first recall some results which will be used throughout this
work.
PROPOSITION 2 (see [4]).
(i) A verifies the growth condition
|A(x, ξ )|

c0 (h0 (x)|ξ | + h1 (x)|ξ |p )

for all ξ ∈ ‫ޒ‬N , a.e. x ∈ .
(ii) A(x, ξ ) is convex with respect to ξ . Moreover, by (A3 ) for all u, v ∈ H we have
u+v
2

1
1
(u) +
(v) − k0 u − v
2
2



(ii) The functionals and I are continuous on H.
(iii) Functional λ is weakly continuously differentiable on H and we have
D

λ (u)(ϕ)

=

a(x, ∇u)∇ϕ dx −λ

f (x, u)ϕ dx

for all u, ϕ ∈ H.
3. Proofs of the theorems.
Proof of Theorem 1. Let us denote by S the best constant in the Sobolev embedding
) → Lp ( ), i.e.

1,p
W0 (

1
p

|∇u|p dx
S=

inf

1,p


Hence, choosing λ = k1 S/C, where S is given by (3.1), we conclude the proof.
We will prove Theorem 2 by using critical point theory. Set f (x, t) = 0 for all t < 0
and consider the energy functional λ : H → ‫ ޒ‬which is given by (1.5).
LEMMA 2. If u is a critical point of

λ

then u is non-negative in

Proof. Observe that if u is a critical point of
of u, i.e. u− (x) = min {u(x), 0} we have
0=D

−
λ (u)(u )

λ,

denoting by u− the negative part

a(x, ∇u)∇u− dx −λ

=

.

f (x, u)u− dx
(3.3)


for all t ∈ ‫ ޒ‬and a.e. x ∈
λ (u)

k1 S p
|t| + Cλ
2p

(3.4)

. Hence,

=

A(x, ∇u) dx −λ

F(x, u) dx

k1
h1 (x)|∇u|p dx −
p
k1
p
u H − Cλ | |,
2p

k1 S p
|u| + Cλ dx
2p

(3.5)

m → ∞.
¨
inequality we deduce that
Using condition (F1 ) combined with Holder’s
0

|f (x, um )||um − u| dx

C
C

|um |p−1 |um − u| dx
p−1
um Lp ( )

um − u

Lp ( )

(3.7)
.

1,p

Since the embedding W0 ( ) → Lp ( ) is compact, {um } converges strongly to u in
Lp ( ). Therefore, relation (3.7) implies that
lim DI(um )(um − u) = 0.

m→∞


(um ))

(see Proposition 2(ii)) implies

lim D (um )(u − um ) = 0.

m→∞

(3.10)

Combining this with Proposition 3(i), we have
(um ) =

lim

m→∞

(u).

(3.11)

We now assume by contradiction that {um } does not converge strongly to u in H,
and then there exist a constant > 0 and a subsequence {umk } of {um } such that
. Using Proposition 2(ii) we get
umk − u H
umk + u
2

1
1


(3.13)
1,p

also converges weakly to u in W0 ( ). So, using

lim inf
k→∞

umk + u
,
2

(3.14)

which contradicts (3.13). Therefore, {um } converges strongly to u in H.
LEMMA 4. There exists a positive constant λ such that for all λ
and hence u1 ≡ 0, i.e. the solution u1 is not trivial.

λ, inf H

λ

< 0,

Proof. Let 0 ⊂ be a compact subset large enough and a function ϕ0 ∈ C0∞ ( )
such that ϕ0 (x) = t1 in 0 and 0 ϕ0 (x) t1 in \ 0 , where t1 is chosen as in
assumption (F2 ): then we have
p



λ and set
⎧
⎪
⎨0,
f (x, t) = f (x, t)
⎪
⎩
f (x, u1 (x))

and F(x, t) =

t
0

for t < 0,
for 0 t u1 (x),
for t > u1 (x),

f (x, s) ds. Define the functional
λ (u)

=

λ

A(x, ∇u) dx −λ

: H → ‫ ޒ‬by
F(x, u) dx .

0 = (D
=

λ (u)

λ,

λ (u1 ))((u

then u

u1 . So, u is a solution of problem

if u is a critical point of

λ

then u

0 as

− u1 )+ )

(a(x, ∇u) − a(x, ∇u1 )) · ∇(u − u1 )+ dx
−λ

=

−D



λ (u)

Proof. We set u = {x ∈ : u(x) > min {u1 (x), t0 }}, where t0 is given as in (F2 ).
Then, by (3.16) and condition (F1 ), we have F(x, u(x)) 0 on \ u . Hence,
λ (u)

k1 u

p
H

−λ

F(x, u) dx .
u

(3.19)


MULTIPLICITY RESULT FOR A CLASS OF p-LAPLACIAN-TYPE EQUATIONS

523

¨
Using (F1 ), Holder’s
inequality and Remark 1(ii), we get
F(x, u) dx

|u|p dx

Therefore,
λ (u)

u

p
H

k1 − λC|

.

(3.21)

In order to prove Lemma 6, it is enough to show that | u | → 0 as u H → 0. Indeed,
⊂ a compact subset, large enough such that
let > 0 be arbitrary; we choose
| \ | < , and denote by u, := u ∩ . Then it is clear that
u

p
H

u

|u|p dx

p

up dx


λ (u),

(3.23)

where := {γ ∈ C([0, 1], H) : γ (0) = 0, γ (1) = u1 }, Lemmas 6–7 show that all of the
assumptions of Proposition 1 are fulfilled, λ (u1 ) = λ (u1 ) < 0 and u1 H > ρ. Then,
c > 0 is a critical value of λ , i.e. there exists u2 ∈ H such that D λ (u2 )(ϕ) = 0 for
all ϕ ∈ H and λ (u2 ) = c > 0. By Lemma 5, 0 u2 u1 in . Therefore, using (3.16)
some simple computations give us
λ (u2 )

=

λ (u2 ),

D

λ (u2 )(ϕ)

=D

λ (u2 )(ϕ)

for all ϕ ∈ H.

By Remark 1(iii), we conclude that u2 is a weak solution of problem (1.1)–(1.2).
Furthermore, λ (u2 ) = c > 0 > λ (u1 ). Thus, u2 is not trivial and u2 = u1 . The proof
of Theorem 2 is now complete.
REFERENCES

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problems, Electron. J. Differential Equations 7 (2003), 1–5.
11. M. Struwe, Variational Methods: Applications to nonlinear partial differential equations
and Hamiltonian systems, 4 ed. (Springer-Verlag, Berlin, 2008).
ˆ Multiplicity of weak solutions for a class of non-uniformly
12. H. Q. Toan and Q.-A. Ngo,
elliptic equations of p-Laplacian type, Nonlinear Anal. 70 (2009), 1536–1546.
13. N. T. Vu, Mountain pass theorem and non-uniformly elliptic equations, Vietnam J.
Math 33 (4) (2005), 391–408.