Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 35, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR THE P-LAPLACIAN
INVOLVING A RADON MEASURE
NEDRA BELHAJ RHOUMA, WAHID SAYEB
Abstract. In this article we study the existence of solutions to eigenvalue
problem
− div(|∇u|p−2 ∇u) − λ|u|p−2 uµ = f
u=0
in Ω,
on ∂Ω
where Ω is a bounded domain in RN and µ is a nonnegative Radon measure.
1. Introduction
In this article study the existence of weak solutions of the quasilinear elliptic
problem
−∆p u − λ|u|p−2 uµ = f (x), in Ω,
(1.1)
u = 0 on ∂Ω,
where Ω is a smooth bounded domain in RN , N ≥ 2, 1 < p < N , and µ is a
nonnegative bounded measure on Ω.
Singular nonlinear problems were studied in [8, 9, 12, 17, 21, 28, 31, 32, 33].
Some recent papers [2, 3, 6, 10, 11, 16, 18] studied functional
Z
Z
Z
1
λ
|u|p
f (x)u(x)dx,
|∇u|p dx −
dx
−
p Ω
p Ω |x|p
Ω
0
where f belongs to Lp (Ω) and λ is a real positive number sufficiently small. This
functional is coercive, and one can expect that there exists a global minimum. Since
1,p
p
the Nemitski operator u(x) 7→ u(x)
|x| from W0 (Ω) in L (Ω) is continuous but not
compact, it is not clear if we can obtain directly the weak lower semicontinuity
of the functional on W01,p (Ω) by using De Giorgi Theorem [13, 19], so that it
seems that we cannot apply the direct methods of the calculus of variations. In
[3], using a critical point technique based on the coercivity and the homogeneity
of the functional, it is shown the existence of a global minimum (for λ belonging
to the set in which the functional is coercive) without using the direct methods
of the calculus of variations. Reference [6] treats more general problems with an
interesting nonvariational method, which does not require homogeneity, but only
2000 Mathematics Subject Classification. 34B15, 34B18, 35A01, 35A02.
Key words and phrases. Dirichlet problem; p-Laplacian; genus function; eigenfunction;
nonlinear eigenvalue problem; Palais-Smale condition; mountain-pass theorem; critical point.
c
2012
Texas State University - San Marcos.
Submitted June 11, 2011. Published February 29, 2012.
1
2
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
coercivity of the quadratic form associated to the equation. In any case, both papers
leave open the question of whether the functional is weakly lower semicontinuous.
In [25], the author proved that the functionals
Z
Z
|u|p
λ
1
p
|∇u| dx −
dx
Hλ (u) =
p Ω
p Ω |x|p
and
Sλ (u) =
1
p
Z
|∇u|p dx −
Ω
λ
p
Z
?
|u|p dx
p/p∗
Ω
W01,p (Ω),
are weakly lower semicontinuous in
provided λ belongs to the set of R
in which the functionals are coercive. Note that both functionals have a nonlinear
term which is continuous but not compact on W01,p (Ω). The author showed the
following result.
0
Theorem 1.1. For all λ ∈ [0, 1/C] and all f ∈ Lp (Ω), 1 < p < N , the problem
|u|p−2 u
+ f (x),
|x|p
u = 0 on ∂Ω,
−∆p u = λ
in Ω,
has a weak solution u ∈ W01,p (Ω), where C = ( N p−p )p is the best constant satisfying
Z
Z
|u|p
dx
≤
C
|∇u|p dx.
p
Ω
Ω |x|
For p = 2, Dupaigne [14] showed that the problem
c
−∆u − 2 u = f in Ω,
|x|
u = 0 on ∂Ω,
has a unique solution for all 0 < c < (p − 2)2 /4 and f ∈ H −1 (Ω). Moreover Peral
[27] showed that the problem
−∆p u −
λ
|u|p−2 u = f
|x|p
u = 0 on ∂Ω,
in Ω,
0
has at least one solution in W01,p (Ω) for all 0 < λ < C1 and f ∈ W −1,p (Ω). For
the proof of this result, the author used the convergence Theorem by Boccardo and
Murat [7].
Remark 1.2. When p > 2, the uniqueness is in general not true, see [15]. However,
the uniqueness in the case 1 < p < 2 seems to be an open problem.
In this paper we assume that the measure µ is a nonnegative Radon measure
satisfying the following assumptions.
(H0) For each Borel set A ⊂ Ω, µ(A) = 0 implies |A| = 0, where | · | denotes the
Lebesgue measure.
(H1) There exists a constant C > 0 such that
Z
Z
p
|u| dµ ≤ C
|∇u|p dx, ∀u ∈ C0∞ (Ω).
Ω
Ω
EJDE-2012/35
EXISTENCE OF SOLUTIONS
3
(H2) There exists (µn )n ⊂ M(Ω) such that for each integer n, the embedding
W01,p (Ω, dx) ,→ Lp (Ω, µn ) is compact, where M(Ω) is the set of bounded
Radon measures.
R
R
(H3) µn % µ in M(Ω); i.e., Ω ϕdµn → Ω ϕdµ, for all ϕ ∈ C0∞ (Ω).
Remark 1.3. When dµ(x) = (1/|x|p )dx, (H1) is the classical Hardy inequality for
p > 1, where the constant C = ( N p−p )p is optimal.
1
Remark 1.4. Let dµ(x) = (δ(x))
p dx, where δ(x) is the distance function to the
boundary, the following inequality holds true (see [20, 26]).
Z
Z
|u|p
dx
≤
C
(Ω)
|∇u|p dx, ∀u ∈ C0∞ (Ω).
n,p
p
Ω
Ω (δ(x))
Moreover, we will show that dµ(x) = |x|1 p dx and dµ(x) =
of measures satisfying (H2) and (H3).
Theorem 1.5. The measure dµ(x) =
tions (H2) and (H3).
1
|x|p dx
1
(δ(x))p dx
and dµ(x) =
are special cases
1
(δ(x))p dx
satisfy condi-
We define the problem
− div(|∇u|p−2 ∇u) − λ|u|p−2 uµn = f
u=0
in Ω,
(1.2)
on ∂Ω.
0
Let f ∈ Lp (Ω). We shall say that u ∈ W01,p (Ω) is a weak solution of (1.2) (resp.
(1.1)) if u satisfies
Z
Z
Z
|∇u|p−2 ∇u∇ϕdx − λ
|u|p−2 uϕdµn =
f ϕdx,
(1.3)
Ω
Ω
Ω
respectively,
Z
Z
Z
|∇u|p−2 ∇u∇ϕdx − λ
|u|p−2 uϕdµ =
f ϕdx,
Ω
Ω
∀ϕ ∈ W01,p (Ω).
(1.4)
Ω
R
Notice that assumption (H3) ensures that the integral Ω |u|p−2 uϕdµ makes sense
whenever u and ϕ are in W01,p (Ω). We prove the following results.
0
Theorem 1.6. Let f ∈ Lp (Ω), 1 < p < N and C satisfying (H1). Then for all
0 < λ < C1 , the problem (1.1) has at least a weak solution u ∈ W01,p (Ω).
Theorem 1.7. Consider the Dirichlet problem
−∆p u − λ|u|p−2 uµ = |u|α−2 u
u=0
in W01,p (Ω),
(1.5)
on ∂(Ω).
For every 0 < λ < 1/C and p < α < p? = N p/(N − p), there exists a nontrivial
solution u ∈ W01,p (Ω).
Note that problem (1.5) has been studied by Peral [27] when dµ(x) =
Next, we prove an auxiliary result.
dx
|x|p .
Theorem 1.8. For every n ∈ N, the problem
−∆p u = λ|u|p−2 uµn
u=0
on ∂Ω.
in Ω,
(1.6)
4
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
has a sequence of eigenvalues (λk )k∈N , such that limk→∞ λk = +∞. Moreover, the
first eigenvalue λ1 (n) is simple, isolated and is defined by
Z
n
o
λ1 (n) = inf k∇ukpLp : u ∈ W01,p (Ω) and
|u|p dµn = 1 .
(1.7)
Ω
Notation. for p > 1, we denote by p0 the real number satisfying
usual W 1,p (Ω) is the Sobolev space equipped with the norm
1/p
kukW 1,p (Ω) = kukpLp + k∇ukpLp
;
1
p
+
1
p0
= 1. As
W01,p (Ω) is the Sobolev space equipped with the norm
1/p
kuk = kukW 1,p (Ω) = k∇ukpLp
.
0
For a positive Radon measure, we set
Z
p
L (Ω, µ) = {u : u is measurable and
|u|p dµ < ∞}.
Ω
When dµ = dx, we set Lp = Lp (Ω, dx).
1
Proof of Theorem 1.5. We start by proving that dµ = |x|1 p dx and dµ = (δ(x))
p dx
?
satisfy conditions (H2) and (H3). For each n ∈ N , we define wn (x) = min(n, |x|−p )
and dµn (x) = wn (x)dx.
Case dµ(x) = |x|1 p dx. Since dµn (x) ≤ ndx, (H2) is obvious. To prove (H3), let
f ∈ C0∞ (Ω). Using the fact that p < N , we obtain
Z
Z
1
f
dx ≤ kf k∞
dx < +∞.
p
p
|x|
|x|
Ω
Ω
On the other hand, since (f wn )n converges to f. |x|1 p , by Dominated Convergence
Theorem we obtain,
Z
Z
f
dx.
lim
f wn dx =
p
n→∞ Ω
Ω |x|
1
?
−p
Case: dµ(x) = (δ(x))
) and
p dx. For n ∈ N , we define wn (x) = min(n, (δ(x))
dµn (x) = wn (x)dx. As in the example from above, (H2) is obvious. Now using the
|f |
fact that |f wn | ≤ (δ(x))
p and
Z
Z
f
dx
≤
C
|∇f |p dx, ∀f ∈ C0∞ (Ω),
p
Ω
Ω (δ(x))
we obtain (H3).
2. Proof of Theorem 1.8
The proof is rather straightforward adaptation of [20, Theorem 3.2] with dµn =
v(x)dx, where the weight v is in Lr with r = r(p, N ) satisfying the following
conditions

N

> p if 1 < p < N,
r = 1 if p > N,


> p if p = N.
Let n ∈ N fixed and define G : W01,p (Ω) → R and F : W01,p (Ω) → R by
Z
Z
1
1
p
|∇u| dx, F (u) =
|u|p dµn .
G(u) =
p Ω
p Ω
EJDE-2012/35
EXISTENCE OF SOLUTIONS
5
In the sequel we consider the functional
φ :W01,p (Ω)→R
7→(G(u))2 − F (u).
u
Proposition 2.1. The functionals G and F are of class C 1 on W01,p (Ω). Moreover
Z
hDG(u), vi =
|∇u|p−2 ∇u∇vdx,
Ω
and
Z
hDF (u), vi =
|u|p−2 uvdµn ,
∀v ∈ W01,p (Ω).
Ω
Proof. We only consider F , the proof for G is similar. Let u and ϕ ∈ W01,p (Ω).
lim
t→0+
1 d
F (u + tϕ) − F (u)
=
F (u + tϕ)|t=0
t
p dt
Z
1 d
=
|u + tϕ|p |t=0 dµn
p dt Ω
Z
1
∂
=
|u + tϕ|p |t=0 dµn
p Ω ∂t
Z
=
|u|p−2 uϕdµn = hDF (u), ϕi.
Ω
The differentiation under the integral is allowed since, if |t| < 1 then
ku + tϕ|p−2 (u + tϕ)ϕ| ≤ (|u| + |tkϕ|)p−1 |ϕ|
≤ (|u| + |ϕ|)p−1 |ϕ| ∈ L1 (Ω, µn ).
Next, we show that DF (u) is continuous. Indeed, by Hölder inequality and using
hypotheses (H1)–(H3), we obtain
Z
|hDF (u), ϕi| = |
|u|p−2 uϕdµn |
Z Ω
p
≤
|u|p−1 |ϕ|dµn ≤ kukp−1
Lp (Ω,µn ) kϕkL (Ω,µn )
Ω
≤ Ckukp−1 kϕk.
Lemma 2.2. The eigenvalues and eigenfunctions associated to the problem (1.6)
are entirely determined by the nontrivial critical values of φ.
Proof. Let u 6≡ 0 be a critical point of φ associated with a critical value c, which
means that φ(u) = c and φ0 (u) = 0. Hence
2G0 (u)G(u) = F 0 (u).
With the condition G0 (u) 6= 0 we obtain G(u) =
F (u) =
1
2λ
= λF (u) thus
1
2λ2
so c = −G2 (u). Therefore,
1
hG0 (u), vi = √ hF 0 (u), vi for all v ∈ Cc∞ (Ω),
2 −c
6
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
1
hφ0 (u), vi = √ hF 0 (u), vi for all v ∈ Cc∞ (Ω).
2 −c
√
Thus, we deduce that λ = 1/(2 −c) is a positive eigenvalue of (1.6) and u is its
associated eigenfunction. Conversely, let (u 6≡ 0, λ) be a solution of (1.6). Then,
for every β ∈ R? , βu is also an eigenfunction associated to λ. In particular for
1/p
β = 1/(2λG(u))
, the function v = (2λG(u))−1/p u is an eigenfunction associated
√
to λ = 1/(2 −c), which proves that v is a critical point associated to the critical
value c = −1/(4λ2 ).
Next, we recall the Genus function defined as follows γ : Σ → N ∪ {∞}, where
Σ = {A ⊂ W01,p (Ω) : A is closed , A = −A} by
γ(A) = min{i ∈ N : ∃ ϕ ∈ C(A, Ri \{0}), ϕ(x) = −ϕ(−x)}.
Let us now consider the sequence
ck = inf sup φ(v),
K∈Ak v∈K
(2.1)
where for k ≥ 1, and
Ak = {K ⊂ W01,p (Ω) : K is compact symmetric and γ(K) ≥ k}.
Proposition 2.3. The values ck defined by (2.1) are the critical values of φ. Moreover ck < 0 for k ≥ 1 and limk→∞ ck = 0.
Proof. The proof is based on the fundamental theorem of multiplicity and the
approximation of Sobolev imbedding by operators of finite rank. We first show
that for all k ≥ 1, ck is a critical value of φ and ck < 0. Since φ is even and is C 1
on W01,p (Ω), then the result follows from the fundamental theorem of multiplicity
if φ satisfies the following conditions:
(1) φ is bounded below.
(2) φ verify the Palais-Smale condition (P-S).
(3) For all k ≥ 1, there exists a compact symmetric subset K such that γ(K) =
k and supv∈K φ(v) < 0.
Let us verify assertion (1). Indeed, condition (H1) implies that
φ(u) ≥
1
kukp (kukp − Cp),
p2
∀u ∈ W01,p (Ω),
which proves that φ is bounded below and φ(u) → +∞ as kuk → +∞.
Assertion (2). We show that φ verify the Palais-Smale condition. Let (uk )k
be a sequence in W01,p (Ω) such that (φ(uk ))k is bounded and (φ0 (uk ))k → 0 in
(W01,p (Ω))0 . Since φ is coercive then (uk )k is bounded in W01,p (Ω). Thus, there
exists a subsequence still denoted by (uk )k such that (∇uk )k converges to ∇u
weakly in Lp , and (uk )k converges to strongly in Lp . By (H3), we obtain that (uk )k
converges strongly in Lp (Ω, µn ). Suppose that (kuk k)k converges to some constant
α ≥ 0. We distinguish two cases.
Case 1: α = 0. Since (uk )k * u in W01,p (Ω) and kuk k → 0, then (uk )k → 0 in
1,p
W0 (Ω). Consequently, the condition (P-S) is satisfied.
Case 2: α > 0. For k ≥ 1 we have
φ0 (uk ) = 2G(uk )G0 (uk ) − F 0 (uk )
EJDE-2012/35
EXISTENCE OF SOLUTIONS
7
which yields
G0 (uk ) =
1
(φ0 (uk ) + F 0 (uk ));
2G(uk )
i.e.,
p (φ0 (uk ) + F 0 (uk ))
= G0 (uk ).
2
k∇uk kpLp
Since u 7→ |u|p−2 u is strongly continuous in Lp (Ω, µn ), kuk k → α > 0 and
(φ0 (uk ))k → 0, then the expression
Vk =
p (φ0 (uk ) + F 0 (uk ))
2
k∇uk kpLp
converges strongly in (W01,p (Ω))0 . However, G0 is continuous, thus uk = (G0 )−1 Vk
converge strongly in W01,p (Ω), from where the (P-S) condition holds.
Next, we prove (3). Indeed, by (H0), there exists a family of balls (Bi )1≤i≤k in
Ω such that Bi ∩ Bj = ∅ if i 6= j and µn (Ω ∩ Bi ) 6= 0. We define
(
−1/p
R
ui (x) Bi |ui |p dµn
if x ∈ Bi ,
vi =
0
if x ∈ Ω\Bi .
Let Xk denote subspace of W01,p (Ω) spanned by {v1 , v2 , . . . , vk }. Since the vi ’s are
Pi=k
linearly independent, we have that dim Xk = k. For each v ∈ Xk , v = i=1 αi vi ,
Pi=k
we obtain F (v) = i=1 |αi |p .
Thus u 7→ (F (u))1/p defines a norm on Xk . Then there exists c > 0 such that
cF (u) ≤ G(u) ≤
1
F (u) ∀u ∈ Xk .
c
Let K be defined as
c2
c2
≤ F (u) ≤ }.
3
2
It is clear that K1 = K ∩ Xk 6= ∅ and supu∈K1 φ(u) < −c/12 < 0. Since Xk is
isomorphic to Rk , one can identify K1 to a crown K10 of Rk such that S k−1 ⊂
K10 ⊂ Rk \{0} where S k−1 is the unit sphere of Rk . Then γ(K1 ) = k and the result
follows.
Finally, we shall prove that limk→+∞ ck = 0. Consider {Ei } sequence of linear
subspaces in W01,p (Ω), such that
• Ei ⊂ Ei+1 ,
• ∪Ei = W01,p (Ω),
• dim(Ei ) = i.
Define
cek = inf
sup φ(v)
K = {u ∈ W01,p (Ω) such that
K∈Ak v∈K∩E c
i−1
Eic
where
is the linear topological complementary of Ei . Obviously cek ≤ ck < 0.
So, it is sufficient to prove that
lim cek = 0.
k→+∞
Assume, by contradiction, that there exists a constant α < 0 such that cek <
α < 0 for all k ∈ N, then for each k ∈ N, there exists Kk such that cek <
8
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
c
supu∈Kk ∩Ei−1
φ(u) < α and there exists uk ∈ Kk ∩ Ei−1
such that cek < φ(uk ) < α.
c
In this way, φ is bounded, hence for some subsequence still denoted (uk ),
uk * u
in W01,p (Ω),
uk → u
in Lp (Ω, µn ).
Hence φ(u) < α < 0, which is a contradiction with the fact that u ≡ 0 because
c
.
uk ∈ Ei−1
Remark 2.4. It is clear that the sequence (λk )k defined by the formula
λk = 2√1−c → +∞ as k → +∞.
k
Remark 2.5. We consider λk = inf K∈Γk supu∈K G(u), where Γk is define by
{K ⊂ W01,p (Ω)\{0} : K is compact, symmetric γ(K) ≥ k, kukLp (Ω,µn ) = 1}.
Particulary
λ1 (n) = inf k∇ukpLp : u ∈ W01,p (Ω) and kukLp (Ω,µn ) = 1 .
Moreover, using [23, Theorem 4.11], we obtain the following result.
Theorem 2.6. If u ∈ W01,p (Ω) is an eigenfunction of (1.3), then u is continuous
in Ω.
In what follows we will use the so-called Picone’s identity proved in [1]. We recall
it here for completeness.
Theorem 2.7 (Picone’s identity). Let u > 0, v > 0 be two continuous functions
in Ω, differentiable a.e.. Denote
up
up−1
p
|∇v|
−
p
|∇v|p−2 − ∇u∇v,
vp
v p−1
up R(u, v) = |∇u|p − |∇v|p−2 ∇ p−1 ∇v.
v
L(u, v) = |∇u|p + (p − 1)
Then
(i) L(u, v) = R(u, v),
(ii) L(u, v) ≥ 0 a.e.,
(iii) L(u, v) = 0 a.e. in Ω if and only if u = kv for some k ∈ R.
We will show that the first eigenvalue λ1 (n) of (1.6) defined by (1.7) is simple
and isolated, and only eigenfunctions associated with λ1 (n) do not change sign.
Proposition 2.8. The first eigenvalue λ1 (n) is simple.
Proof. Let u, v be two eigenfunctions associated to λ1 (n) and fixed > 0. We can
assume without restriction that u and v are positive in Ω. From Picone’s identity
we have
Z
Z
L(u, v + )dx =
R(u, v + )dx
Ω
Ω
Z
Z
up
p
u dµn −
|∇v|p−2 ∇(
= λ1 (n)
)∇vdx.
(v + )p−1
Ω
Ω
EJDE-2012/35
EXISTENCE OF SOLUTIONS
9
The functional up /(v + )p−1 belongs to W01,p (Ω) and then it is admissible for the
weak formulation of −∆p u = λ1 (n)|u|p−2 uµn . It follows that
Z
Z
v p−1
0≤
L(u, v + )dx = λ1 (n)
)dµn .
up (1 −
(v + )p−1
Ω
Ω
Letting → 0, we obtain L(u, v) = 0, a.e. in Ω, and therefore using (iii), we obtain
u = kv.
Proposition 2.9. Let u ∈ W01,p (Ω) be a nonnegative weak solution of (1.6), then
either u ≡ 0 or u(x) > 0 for all x ∈ Ω.
The proof of the above proposition is a direct consequence of Harnack’s inequality, see [35, 36].
Theorem 2.10. Let (u, λ) ∈ W01,p (Ω) × R+ be an eigensolution of (1.1). Then
u ∈ L∞ (Ω, µn ).
The proof of the above theorem is rather a straightforward adaptation of [22,
Theorem 4.1] with dµn = dx.
Theorem 2.11. Let u be an eigenfunction of (1.6) associated to an eigenvalue
λ 6= λ1 (n) and 1 ≤ q < p. We define
Z
Z
q
p
I = min
|u| dµn , u ∈ L (Ω, µn ),
|u|p dµn = 1 .
Ω
Ω
Then
pq
min(µn (Ω− ), µn (Ω+ )) ≥ ((Cλ)−1/p I) p−q ,
where Ω+ = {x ∈ Ω, u(x) > 0} and Ω− = {x ∈ Ω, u(x) < 0}.
Proof. Let u be an eigenfunction associated to λ, then
Z
Z
p−2
|∇u| ∇u∇vdx = λ
|u|p−2 uvdµn , ∀v ∈ W01,p (Ω).
Ω
(2.2)
(2.3)
Ω
For λ 6= λ1 (n), u changes sign i.e., u+ 6= 0 and u− 6= 0. Since u+ ∈ W01,p (Ω) we
have
Z
Z
|∇u+ |p dx = λ
|u+ |p dµn .
Ω
Ω
For 1 ≤ q < p, we have:
Z
Z
q/p
q
|u+ |q dµn ≤
|u+ |p dµn
(µn (Ω+ ))1− p
Ω
Ω
Z
q/p
q
q/p
≤ C µn (Ω+ )1− p
|∇u+ |p dx
Ω
Z
q/p
q
q/p
+ 1− p
≤ (λC) µn (Ω )
|u+ |p dµn
Ω
and
ku+ kqLq (Ω,µn ) ≤ (λC)q/p µn (Ω+ )
p−q
pp
ku+ kLp (Ω,µn ) .
Finally
pq
µn (Ω+ ) ≥ ((Cλ)−1/p I) p−q .
Now, we establish the isolation of the first eigenvalue.
(2.4)
10
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
Theorem 2.12. Let λ1 (n) be the first eigenvalue of the problem (1.6). Then λ1 (n)
is isolated.
Proof. Our approach is related to the method of [4, 5]. Let λ > 0 be an eigenvalue
of (1.6) and let v be the corresponding eigenfunction. By (1.7), it follows that
λ1 (n) < λ and so λ1 (n) is left-isolated. To prove that λ1 (n) is right-isolated, we
argue by contradiction. We suppose that there exists a sequence of eigenvalues
(λk )k∈N , such that λk 6= λ1 (n) and λk → λ1 (n). Let (uk )k∈N be the corresponding
sequence of eigenfunctions such that
Z
|∇uk |p dx = 1, ∀ k ∈ N.
(2.5)
Ω
There exists a subsequence, denoted again by (uk )k and a function u ∈ W01;p (Ω)
such that
uk * u
uk → u
on W01,p (Ω)
on Lp (Ω, µn ).
Our next aim is to show that u is the eigenfunction corresponding to λ1 (n). First,
0
since −∆p is a continuous and one-to-one operator from W01,p (Ω) into W0−1,p (Ω)
0
and so is its inverse operator (−∆p )−1 defined from W0−1,p (Ω) into W01,p (Ω) (see
[27]). Thus,
uk = (−∆p )−1 (λk |uk |p−2 uµn ).
By Vitali’s Theorem, we have
p
λk up−2
uk → λ|u|p−2 u
k
0
strongly in L p−1 (Ω, µn ) ,→ W −1,p (Ω).
The continuity property of (−∆p )−1 implies that
uk → u
strongly in W01,p (Ω).
Hence, u is an eigenfunction of (1.6), corresponding to λ1 (n). Using Vitali’s Theorem, again, we have
|∇uk |p−2 ∇uk → |∇u|p−2 ∇u strongly in L1 .
Z
p−1
Z
Z
1/p
p
|uk |p dµn
|uk |p−2 uk vdµn ≤
.
|v|p dµn
≤ kvk.
Ω
Ω
Ω
It should appear a constant η > 0 for every > 0 and Ω ⊂ Ω such that
and u(x) ≥ 2η for every x ∈ Ω .
µn (Ω\Ω ) ≤
2
Let us denote
(2.6)
Ω+
k = {x ∈ Ω, uk (x) > 0},
(2.7)
Ω−
k
(2.8)
= {x ∈ Ω, uk (x) < 0}.
Moreover, by Egorov’s Theorem, there exists Ω0 ⊂ Ω such that
µn (Ω\Ω0 ) ≤
2
and uk converges uniformly to u. On the other hand, there exists N > 0 such that
for every k > N , we have
Ω ∩ Ω0 ⊂ Ω+
k
EJDE-2012/35
EXISTENCE OF SOLUTIONS
11
and then
0
0
µn (Ω+
k ) ≥ µn (Ω ∩ Ω ) ≥ µn (Ω) − (µn (Ω\Ω ) + µn (Ω\Ω )) ≥ µn (Ω) − .
pq
−
−1/p
I) p−q . If
Hence, it follows that µn (Ω+
k ) and µn (Ωk ) ≥ K, where K = ((Cλ)
K
we choose = 2 , we obtain
+
µn (Ω) = µn (Ω−
k ) + µn (Ωk ) ≥ µn (Ω) − + K = µn (Ω) + > µn (Ω),
which is a contradiction. Therefore λ1 (n) is isolated.
3. Proof of Theorem 1.6
Lemma 3.1. Let λ1 (n) be the first eigenvalue associated to (1.6). Then, λ1 (n) ≥
and limn→∞ λ1 (n) = C1 .
1
C
Proof. Notice that
R
R
|∇u|p dx
|∇u|p dx
1
Ω
RΩ
λ1 (n) = inf R
≥
inf
≥ .
p
p
Ω
Ω
C
|u| dµn
|u| dµ
Ω
Ω
Since (λ1 (n))n is a non increasing sequence, we have to prove that the limit can
not be larger than C1 . Assume by contradiction that limn→∞ λ1 (n) = C1 + δ, for
some δ > 0. Then we can choose φ ∈ C0∞ (Ω) such that
R
|∇φ|p dx
1
δ
RΩ
<
+ .
p dµ
C
2
|φ|
Ω
Which gives us
R
|∇φ|p dx
δ
1
λ1 (n) ≤ RΩ p
+
≤
C
2
|φ| dµn
Ω
for n large enough.
In the sequel for λ > 0 let us denote by Lµλn the operator defined on W01,p (Ω) by
Lµλn u = −∆p u − λ|u|p−2 uµn .
The first result in this section is an easy consequence of the Hardy’s inequality.
Lemma 3.2. If 0 < λ <
1
C,
then Lµλn is a positive operator.
Proof. From assumption (H1) we have
hLµλn u, ui ≥ (1 − λC)kukp ≥ 0
whenever 0 < λ < 1/C.
Next we recall a formula from [34].
Lemma 3.3. Let a, b ∈ RN and h., .i be the standard scalar product in RN . Then
(
Cp |a − b|p
if p ≥ 2
p−2
p−2
h|a| a − |b| b, (a − b)i ≥
|a−b|2
Cp (|a|+|b|)2−p if 1 < p < 2.
0
Lemma 3.4. The operator Lµλn : W01,p (Ω) → W −1,p (Ω) is uniformly continuous
on bounded sets.
12
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
Proof. Assume p > 2 and consider K ⊂ W01,p (Ω) be a bounded set; i.e., there exists
M > 0 such that
kuk ≤ M, ∀u ∈ K.
Then, using Lemma 3.3 and Hölder inequality, for u, v ∈ K and φ ∈ W01,p (Ω), we
obtain
|hLµλn u − Lµλn v, φi|
Z
Z
p−2
p−2
≤ (|∇u|
+ |∇v| )|∇u − ∇v||∇φ|dx + λ (|u|p−2 + |v|p−2 )|u − vkφ|dµn
Ω
Ω
≤ 2cp M p−2 k∇u − ∇vkLp + 2λcp M p−2 ku − vkLp (Ω,µn )
≤ 2cp M p−2 (min{1, Cλ})ku − vk.
The same process is applied for 1 < p < 2.
Lemma 3.5. The operator
Lµλn
:
W01,p (Ω)
0
→ W −1,p (Ω) is pseudo-monotone.
Proof. Let (uk )k≥1 ⊂ W01,p (Ω) such that uk * u in W01,p (Ω) and
lim sup < Lµλn uk , uk − u >≤ 0.
k→∞
We want to prove that
lim infhLµλn uk , uk − vi ≥ hLµλn u, u − vi for all v ∈ W01,p (Ω).
Since uk * u in W01,p (Ω), it follows that
Z
|∇u|p−2 ∇u∇(uk − u)dx → 0,
as k → +∞.
(3.1)
Ω
We estimate
Z
Ω
p−1
|uk |p−2 uk (uk − u)dµn ≤ kuk kL
p (Ω,µ ) kuk − ukLp (Ω,µn ) .
n
Through uk → u in Lp (Ω, µn ) and uk is bounded in Lp (Ω, µn ) then
Z
|uk |p−2 uk (uk − u)dµn → 0, as k → +∞.
Ω
So
lim sup
Z
k→∞
|∇uk |p−2 ∇uk ∇(uk − u)dx + (−λ
Ω
Z
|uk |p−2 uk (uk − u)dµn ) ≤ 0,
Ω
which yields
Z
lim sup
k→∞
|∇uk |p−2 ∇uk ∇(uk − u)dx ≤ 0.
(3.2)
Ω
Combining (3.1), (3.2) and Lemma 3.3 we obtain that (uk )k → u in W01,p (Ω). The
proof is complete.
Proposition 3.6. For every 0 < λ < 1/C, the operator Lµλn is coercive.
Proof. Using (H1)–(3̋), we have
Z
Z
Z
hLµλn u, ui =
|∇u|p dx − λ
|u|p dµn ≥ (1 − λC)
|∇u|p dx,
Ω
which implies that
Lµλn
Ω
is coercive, whenever 0 < λ <
Ω
1
C.
EJDE-2012/35
EXISTENCE OF SOLUTIONS
13
0
By Proposition 3.6 and Lemma 3.5 the operator Lµλn : W01,p (Ω) → W −1,p (Ω)
is coercive, bounded from below and pseudo-monotone. Hence, by [23, Theorem
4.11], it is onto. Thus we have the following result.
0
Theorem 3.7. For every f ∈ Lp , there exists un ∈ W01,p (Ω) which is a solution
of (1.2).
Lemma 3.8. For each n ∈ N, let un be a solution of the Dirichlet problem (1.2).
Then the sequence (un )n is bounded in W01,p (Ω).
Proof. Since
Z
Z
p
|∇un | dx − λ
Ω
Z
p
|un | dµn =
f un dx,
Ω
Ω
and using (H1), we obtain
(1 − λC)kun kp ≤ kf kLp0 kun kLp ,
1
kf kLp0 p−1
kun k ≤ C1
,
(1 − λC)
where C1 is the positive constant of the continuous of Sobolev embedding satisfied.
Lemma 3.9. Let (un )n be the sequence as defined in Theorem 3.7. Then (un )n
converges to a weak solution u of (1.1).
Proof. By Lemma 3.8, since un is bounded in W01,p (Ω), we have
un * u
in W01,p (Ω),
un * u
in Lp (Ω, µ),
un → u
(3.3)
p
in L .
Then
Z
|∇un |p−2 ∇un ∇ϕdx → |∇u|p−2 ∇u∇ϕdx,
for all ϕ ∈ W01,p (Ω).
Ω
Next, we show that
Z
Z
|un |p−2 un ϕdµn →
|u|p−2 uϕdµ,
Ω
for all ϕ ∈ C0∞ (Ω).
Ω
Indeed, we have
Z
Z
|un |p−2 un ϕdµn −
|u|p−2 uϕdµ
Ω
Ω
Z
Z
p−2
=
|u| uϕd(µ − µn ) − (|un |p−2 un − |u|p−2 u)ϕdµn Ω
Ω
Z
p−1
≤ kukLp (Ω,µ−µn ) kϕkLp (Ω,µ−µn ) + | (|un |p−2 un − |u|p−2 u|)ϕ|dµn |.
Ω
So, by (H3) the first integral converges to 0, as n → ∞; respectively by the weak
convergence in (3.3), the second integral converges to 0, as n → ∞. Therefore,
u is a solution of our problem in the sense of distributions. Moreover by density
argument and taking into account that u ∈ W01,p (Ω), we conclude that u is solution
in the sense of W01,p (Ω).
14
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
4. Proof of Theorem 1.7
Some recent papers [17, 21, 28, 30, 32, 29] considered a class of functionals
with the minimax method. We will use again the variational approach to study
the case of unbounded functionals, more precisely the existence of solution via the
mountain-pass theorem. For instance the following result holds. For 0 < λ < C1 ,
let
1
1
J(u) = hLµλn u, ui − kukα
u ∈ W01,p (Ω).
Lα ,
p
α
To obtain a nontrivial critical point of the functional J, we apply the following
version of the mountain-pass theorem from [24] with the usual Palais-Smale compactness condition. So the critical points of the functional J are a weak solutions
for (1.5).
Theorem 4.1. Let E be a real Banach space and J ∈ C 1 (E, R) satisfying PalaisSmale condition. Suppose that J(0) = 0 and for some σ, ρ > 0 and e ∈ E, with
kek > ρ, one has σ ≤ inf kuk=ρ J(u) and J(e) < 0. Then J has a critical value c ≥ σ
characterized by
c = inf max J(γ(t)),
γ∈Γ t∈[0,1]
where
Γ = {γ ∈ C([0, 1], E) : γ(0) = 0, γ(1) = e}.
The proof of the above theorem follows from the following lemma.
Lemma 4.2. The functional J satisfies the Palais-Smale condition.
Proof. Let (uk )k ∈ W01,p (Ω) be a Palais-Smale sequence. Set
c = lim J(uk ),
k→∞
J 0 (uk ) = k
such that (k )k → 0. Thus
|J 0 (uk )w| ≤ k kwk, for all w ∈ W01,p (Ω).
For k large enough, we will have
1 0
1
hJ (uk ), uk i + hJ 0 (uk ), uk i,
α
α
1
1
1
p
≥ ( − )(1 − λC)kuk k − kuk kk ,
p α
α
1
1
1
p
≥ ( − )(1 − λC)kuk k − kuk k.
p α
α
c + 1 ≥ J(uk ) −
Hence, the sequence (uk )k is bounded in W01,p (Ω). By compactness argument we
can assume that
uk * u
uk → u
in W01,p (Ω),
in Lα (Ω),
for p < α < p? .
Using (H2), we obtain that (uk )k converges to u in Lp (Ω, µn ). It follows that
0
0
|uk |p−2 uk → |u|p−2 u in Lp (Ω, µn ), hence in W −1,p (Ω). Let us denote by Vk =
|uk |p−2 uk µn − |uk |α−2 uk and V = |u|p−2 uµn − |u|α−2 u. Since (−∆p )−1 is continuous, we conclude that
uk = (−∆p )−1 (Vk ) converges to (−∆p )−1 (V ) = u.
Therefore, |uk |p−2 uk converges to |u|p−2 u in W01,p (Ω).
EJDE-2012/35
EXISTENCE OF SOLUTIONS
15
Lemma 4.3. The functional J satisfies the conditions for the mountain-pass theorem.
Proof. Let δ1 = αδ. First, we show that there exist positive constants ρ and α1
such that
J(u) ≥ α, if kuk = ρ,
1,p
and there exists ϕ ∈ W0 (Ω) such that J(tϕ) → −∞, as t → ∞. Indeed, for
u ∈ W01,p (Ω), we have
1
1
1
δ1
p
J(u) = hLµλn u, ui − kukα
kukα .
Lα ≥ (1 − λC)kuk −
p
α
p
α
Since λ < 1/C and p < α, we can set
(1 − λC)αS α/p (1 − λC)α 1/(α−p) 1
1
1/α−p)
−
,
ρ=
,
α
=
α
1
1− p?
p
(δ1 )
p α
|Ω|
such that J(u) ≥ α1 if kuk = ρ.
Let us prove the second assertion. Let t > 0 large enough, and choose ϕ ∈
W01,p (Ω)\{0} satisfying
1
1
J(tϕ) = tp hLµλn ϕ, ϕi − tα kϕkα
Lα → −∞ as t → +∞.
p
α
Thus, we have J(tϕ) < 0, for sufficiently large t.
So, we can conclude that J has a critical value c ≥ α1 , which can be characterized
by
c = inf max J(γ(t)),
γ∈Γ t∈[0,1]
where
Γ = {γ ∈ C([0, 1], W01,p (Ω)), γ(0) = 0, γ(1) = e}.
Next, we shall prove the positivity of the solution. Multiply the equation −∆p u−
λ|u|p−2 uµn = |u|α−2 u by u− and integrate over Ω, we find ku− k = 0 and so u is a
µn
positive solution of (Pα,λ
) the proof is complete.
For the proof of Theorem 1.7, we need the following results.
Lemma 4.4. Let (un )n be a sequence of weak solutions of (1.5) with µn instead of
µ. Then, (un )n is bounded in W01,p (Ω).
Proof. As un is a weak solution of (1.5) with µn instead of µ, then un is a critical
point of the functional J. Since J satisfies the Palais-Smale condition, then (un )n
is bounded in W01,p (Ω).
Lemma 4.5. Let (un )n be a sequence of weak solutions of the problem (1.5) with
µn instead of µ. Then (un )n converges to a weak solutions u of (1.5).
Proof. By Lemma 3.8, since (un )n is bounded in W01,p (Ω), it follows that
un * u
in W01,p (Ω),
un * u
in Lp (Ω, µ),
un → u
α
in L
(4.1)
?
p<α<p .
Hence
Z
Ω
|∇un |p−2 ∇un ∇ϕdx → |∇u|p−2 ∇u∇ϕdx,
for all ϕ ∈ W01,p (Ω).
16
N. BELHAJ RHOUMA, W. SAYEB
EJDE-2012/35
By the compactness of Sobolev embedding, we obtain
Z
Z
|un |α−2 un ϕdx →
|u|α−2 uϕdx, for all ϕ ∈ W01,p (Ω).
Ω
Ω
Next, we show that
Z
Z
|un |p−2 un ϕdµn →
|u|p−2 uϕdµ,
Ω
for all ϕ ∈ C0∞ (Ω).
Ω
Indeed, we have
Z
Z
|un |p−2 un ϕdµn −
|u|p−2 uϕdµ
Ω
Ω
Z
Z
p−2
=
|u| uϕd(µ − µn ) − (|un |p−2 un − |u|p−2 u)ϕdµn Ω
Ω
Z
p−1
≤ kukLp (Ω,µ−µn ) kϕkLp (Ω,µ−µn ) + | (|un |p−2 un − |u|p−2 u|)ϕ|dµn |.
Ω
So, using (H3) the first integral converges to 0, as n → ∞ respectively by the weak
convergence in (4.1), the second integral converges to 0, as n → ∞. Therefore,
u is a solution of our problem in the sense of distribution. Moreover by density
argument and taking into account that u ∈ W01,p (Ω), we conclude that u is solution
in the sense of W01,p (Ω).
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N. BELHAJ RHOUMA, W. SAYEB
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Nedra Belhaj Rhouma
Departement de Mathématiques, Facultédes sciences de Tunis, Université Tunis El Manar, Campus Universitaire 2092, Tunis, Tunisia
E-mail address: nedra.belhajrhouma@fst.rnu.tn
Wahid Sayeb
Departement de Mathématiques, Facultédes sciences de Tunis, Université Tunis El Manar, Campus Universitaire 2092, Tunis, Tunisia
E-mail address: wahid.sayeb@yahoo.fr