Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 95, pp. 1–6.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE RESULTS FOR A P-LAPLACIAN PROBLEM WITH
COMPETING NONLINEARITIES AND NONLINEAR
BOUNDARY CONDITIONS
DIMITRIOS A. KANDILAKIS, MANOLIS MAGIROPOULOS
Abstract. By using the fibering method we study the existence of nonnegative solutions for a class of quasilinear elliptic problems in the presence of
competing subcritical nonlinearities.
1. Introduction
In this paper we study the problem
∆p u = a(x)|u|p−2 u − b(x)|u|q−2 u
in Ω
(1.1)
∂u
|∇u|p−2
= λc(x)|u|p−2 u on ∂Ω,
∂ν
where Ω is a bounded domain in RN with a sufficiently smooth boundary ∂Ω, ν
is the outward unit normal vector on ∂Ω, 1 < q < p < N , a(·), b(·) ∈ L∞ (Ω)
with a(x) > θ > 0, b(x) > 0 a.e., c(x) ∈ L∞ (∂Ω), with c(x) > 0 a.e. As usual,
∆p u = div(|∇u|p−2 ∇u) denotes the p-Laplacian operator.
When b ≡ 0, problem (1.1) appears naturally in the study of the Sobolev trace inequality. Since the embedding W 1,p (Ω) ⊆ Lp (Ω) is compact there exists a constant
λ1 such that
1/p
λ1 kukLp (∂Ω) ≤ kukW 1,p (Ω) .
The functions at which equality holds; that is,
λ1 :=
u∈W
kukpW 1,p (Ω)
inf
1,p
(Ω)\{0}
kukpLp (∂Ω)
,
(1.2)
are called extremals and are the solutions to the problem
∆p u = a(x)|u|p−2 u
in Ω
∂u
= λ1 c(x)|u|p−2 u
|∇u|p−2
∂ν
For more details we refer the reader to [9].
(1.3)
on ∂Ω,
2000 Mathematics Subject Classification. 35J60, 35J92, 35J25.
Key words and phrases. Quasilinear elliptic problems; subcritical nonlinearities;
fibering method.
c
2011
Texas State University - San Marcos.
Submitted July 14, 2010. Published July 28, 2011.
1
2
D. A. KANDILAKIS, M. MAGIROPOULOS
EJDE-2011/95
Problems of the form ∆p u = ±λ|u|p−2 u + f (x, u) with Dirichlet boundary conditions has been extensively studied, see for example [1, 6, 7, 10, 13]. Recently, this
problem with nonlinear boundary conditions has been considered in [3, 4, 12].
In this paper we employ Pohozaev’s fibering method in order to show that if
λ < λ1 , then (1.1) admits a nonnegative solution. In the case λ = λ1 , the fibering
method is no longer applicable, so we introduce the term εd(·)|u|s−2 u in the equation, where ε > 0 and d(·) ∈ L∞ (Ω), d(·) > 0 a.e., and examine the behavior of the
solutions uε as ε → 0. It turns out that kuε kW 1,p (Ω) → +∞ and the energy of the
solutions diverges to −∞.
2. Main results
R
Our reference space is W 1,p (Ω) equipped with the norm kukp = Ω [|∇u|p +
a(x)|u|p ]dx, which is equivalent to its usual one. In what follows, σ(·) is the surface
measure on the boundary of Ω.
The energy functional associated with (1.1) is
Z
Z
Z
1
1
λ
Φλ (u) :=
[|∇u|p dx+a(x)|u|p ]dx−
b(x)|u|q dx−
c(x)|u|p dσ(x). (2.1)
p Ω
q Ω
p ∂Ω
Following [9], let λ1 ∈ R be the first positive eigenvalue of (1.3), given by (1.2).
Theorem 2.1. Suppose that 1 < q < p < N and λ < λ1 . Then (1.1) admits a
nonnegative solution.
Proof. We employ the fibering method introduced in [11], see also [2] and [8], in
order to prove the existence of a negative energy solution of (1.1). Writing u = rv,
r > 0 and v ∈ W 1,p (Ω), we have
Z
Z
Z
rp
rp
rq
Φλ (rv) =
|∇v|p dx +
a(x)|v|p dx −
b(x)|v|q dx
p Ω
p Ω
q Ω
Z
(2.2)
λrp
p
−
c(x)|v| dσ(x).
p ∂Ω
For u 6= 0 to be a critical point, it should hold ∂Φλ∂r(rv) = 0, from which we obtain
Z
Z
Z
rp−q
|∇v|p dx + rp−q
a(x)|v|p dx − λrp−q
c(x)|v|p dσ(x)
Ω
Ω
∂Ω
Z
(2.3)
q
=
b(x)|v| dx,
Ω
ensuring the existence of a unique r = r(v) > 0 satisfying (2.3). By the implicit
function theorem [14, Thm. 4.B], the function v → r(v) is continuously differentiable for v 6= 0. Notice that
r(kv)kv = r(v)v
for k > 0.
(2.4)
In view of (2.2) and (2.3),
Φλ (r(v)v) =
1 1
− r(v)q
p q
Z
b(x)|v|q dx < 0.
Ω
Consider the functional
Z
Z
Z
p
p
H(u) :=
|∇u| dx +
a(x)|u| dx − λ
Ω
Ω
∂Ω
c(x)|u|p dσ(x).
EJDE-2011/95
P-LAPLACIAN PROBLEM WITH COMPETING NONLINEARITIES
3
By the way we chose λ, for u ∈ W 1,p (Ω), H(u) ≥ 0 (equality holds exactly when
u = 0). Define V = {v ∈ W 1,p (Ω) : H(v) = 1}. Evidently, (H 0 (v), v) 6= 0 for v ∈ V .
b λ (v) = Φλ (r(v)v)
In view of [2, Lemma 3.4], any conditional critical point of Φ
subject to H(v) = 1, provides a critical point r(v)v of Φλ . Notice that V is
bounded. To see this, let ε > 0 be such that λ + ε < λ1 . Then, for v ∈ V , by the
definition of λ1 ,
R
R
|∇v|p + Ω a|v|p
Ω
λ+ε< R
,
c(x)|v|p dσ(x)
∂Ω
which implies that
Z
Z
Z
Z
p
p
p
1 = |∇v| + a|v| − λ
c(x)|v| dσ(x) > ε
c(x)|v|p dσ(x).
Ω
Ω
∂Ω
∂Ω
R
Thus ∂Ω c(x)|v|p dσ(x), v ∈ V , is bounded. Consequently, V is a bounded set. Because of the embedding W 1,p (Ω) ,→ Lq (Ω), (2.3) guarantees that r(V ) is bounded.
Consequently, I = {Φλ,µ (r(v)v) : v ∈ V } is a bounded interval in R with endpoints
a and b, a < b ≤ 0. We are now going to show that a ∈ I. To this end, let {vn }n∈N
be a sequence in V , with Φλ (r(vn )vn ) → a. Without loss of generality, we may
assume that vn → v weakly in W 1,p (Ω). We may also assume that r(vn ) → r ∈ R.
Thus r(vn )vn → rv weakly in W 1,p (Ω). Since Φλ (·) is weakly lower semicontinuous,
Φλ (rv) ≤ lim inf Φλ (r(vn )vn ) = a,
n→∞
(2.5)
ensuring that rv 6= 0. Because of the compactness of the Sobolev and trace embeddings, r(vn )vn → rv strongly in Lq (Ω), Lp (∂Ω), respectively. Taking into account
the lower semicontinuity of the norm in (2.3), we have
Z
p−q
r
H(v) ≤
b(x)|v|q dx.
(2.6)
Ω
Combining (2.3) and (2.6), we get r ≤ r(v). Our purpose is to prove equality. Let
us assume the contrary; that is r < r(v). We define F (y) = Φλ (yv), y ≥ 0. For
y ∈ [r, r(v)], we have
Z
0
q−1
p−q
F (y) = y
y
H(v) −
b(x)|v|q dx ,
(2.7)
Ω
which is negative everywhere, but at y = r(v). Thus F (y) decreases strictly in the
considered interval, giving
Φλ (r(v)v) < Φλ (rv) ≤ a,
(2.8)
because of (2.5). Notice that for suitable k ≥ 1, kv ∈ V . Then, combining (2.4)
and (2.8), we obtain
Φλ (r(kv)kv) = Φλ (r(v)v) < Φλ (rv) ≤ a,
which is a contradiction. So, r = r(v), and necessarily Φλ (r(kv)kv) = a. This
b λ (·) subject to H(v) = 1, and,
means that kv is a conditional critical point of Φ
consequently, r(kv)kv = r(v)v is a critical point of Φλ (·). Since for a minimizer w,
|w| is also a minimizer, we may assume v ≥ 0, and r(v)v is a nontrivial nonnegative
solution of (1.1).
In attempting to obtain the existence of a solution to problem (1.1) for λ = λ1 ,
following a similar procedure, we encounter an unsurpassable difficulty, due to the
fact that (2.3) does no longer guarantee the existence of a suitable r(v). In order to
4
D. A. KANDILAKIS, M. MAGIROPOULOS
EJDE-2011/95
study this situation, we add an additional term in (1.1), with the problem taking
the following form
∆p u = a(x)|u|p−2 u − b(x)|u|q−2 u + εd(x)|u|s−2 u
in Ω,
(2.9)
∂u
= λ1 c(x)|u|p−2 u on ∂Ω,
∂ν
where, ε > 0, q < s < p∗ , and d(·) ∈ L∞ (Ω) with d(·) > 0 a.e. in Ω. The energy
functional is
ε
Fλ1 ,ε (u) := Φλ1 (u) + D(u),
(2.10)
s
where
Z
D(u) :=
d(x)|u|s dx.
|∇u|p−2
Ω
Theorem 2.2. Suppose that 1 < q < s < p∗ , ε > 0 and d(·) ∈ L∞ (Ω) with d(·) > 0
a.e. in Ω.Then problem (2.9) admits a nonnegative solution uε for every ε > 0.
Furthermore, Fλ1 ,ε (uε ) → −∞ and kuε k → +∞ as ε → 0.
Proof. Following a similar reasoning, we obtain the counterpart of (2.3),
Z
Z
Z
p−q
p
p
r
|∇v| dx +
a(x)|v| dx − λ1
c(x)|v|p dσ(x)
Ω
Ω
∂Ω
Z
s−q
s
+ εr
d(x)|v| dx
Ω
Z
=
b(x)|v|q dx.
(2.11)
Ω
The function R(y) = Hy p−q + εDy s−q − B, with H ≥ 0, D, B > 0, has a unique
root in (0, +∞), since it is strictly increasing, R(0) = −B and R(y) → +∞, for
y → +∞. Thus, for v ∈ W 1,p (Ω) there exists a unique positive rε (v) satisfying
(2.11). The so defined function v → rε (v) is once more continuously differentiable
for v 6= 0, by another application of the implicit function theorem. In addition, it is
easily checked that (2.4) remains true. We notice also that, due to (2.11), if v 6= 0,
1 1
1 1
(2.12)
− rε (v)p H(v) + ε − rε (v)s D(v) < 0.
Fλ1 ,ε (rε (v)v) =
p q
s q
We define next the positive functional (except at u = 0),
L(u) := H(u) + D(u).
(2.13)
Consider the set
W = {v ∈ W 1,p (Ω) : L(v) = 1}.
Because of our hypothesis on d(·), (L0 (v), v) > D(v) > 0 for v ∈ W . As usual, the
conditional critical points of Fbλ1 ,ε (v) = Fλ1 ,ε (rε (v)v) subject to L(v) = 1 provide
critical points rε (v)v of Fλ1 . We claim that W is bounded. Indeed, if not, there
would exist vn ∈ W, n ∈ N, such that kvn k → +∞. Let vn := tn un with tn > 0 and
kun k = 1. Since un , n ∈ N, is bounded, by passing to a subsequence if necessary,
we may assume that un → u0 weakly in W 1,p (Ω) and strongly in Lp (c, ∂Ω) and
Ls (Ω). By (2.13),
Z
Z
tpn
|∇un |p dx +
a(x)|un |p dx
Ω
Ω
Z
Z
p
s
− λ1
c(x)|un | dσ(x) + tn
d(x)|un |s dx = 1,
∂Ω
Ω
EJDE-2011/95
P-LAPLACIAN PROBLEM WITH COMPETING NONLINEARITIES
5
and so
Z
0≤
p
Z
|∇un | dx +
Ω
p
Z
a(x)|un | dx − λ1
Ω
c(x)|un |p dσ(x) ≤
∂Ω
and
Z
1
→0
tpn
1
→ 0.
tsn
By (2.15), u0 = 0. On the other hand, since kun k = 1, (2.14) yields
Z
λ1
c(x)|u0 |p dσ(x) = 1
d(x)|un |s dx ≤
0<
(2.14)
(2.15)
Ω
∂Ω
and so u0 6= 0, a contradiction, thereby proving the claim. We can now continue
as in the previous case. Namely, we notice that by the way it was defined, rε (v)
is bounded on W (we use now the embedding W 1,p (Ω) ,→ Lq (Ω)). Thus I 0 =
{Fλ1 ,ε (rε (v)v) : v ∈ W } is a bounded interval with endpoints a0 and b0 , a0 < b0 ≤ 0.
Let {vn }n∈N be a sequence of W with Fλ1 ,ε (rε (vn )vn ) → a0 . We may assume that
vn → vε weakly in W 1,p (Ω), and rε (vn ) → rε ∈ R. Thus rε (vn )vn → rε vε weakly
in W 1,p (Ω) , and consequently, at least for a subsequence, strongly in Ls (Ω). Since
Φλ1 (·) is weakly lower semicontinuous, so is Fλ1 ,ε (·), and the obvious counterpart of
(2.5) ensures that rε vε 6= 0. Combining (2.11) with the lower semicontinuity of the
involved norms, the compactness of the Sobolev and trace embeddings W 1,p (Ω) ,→
Lq (Ω), W 1,p (Ω) ,→ Ls (Ω), and W 1,p (Ω) ,→ Lp (∂Ω), respectively, we obtain
Z
p−q
s−q
rε H(vε ) + rε εD(vε ) ≤
b(x)|vε |q dx = B(vε ).
(2.16)
Ω
Evidently, (2.11) and (2.16) ensure that rε ≤ rε (vε ). We are going to prove equality.
Assuming the contrary, the function G(y) = Fλ1 ,ε (yvε ), y > 0, has its derivative
G0 (y) = y q−1 y p−q H(vε ) + y s−q εD(vε ) − B(vε )
which is negative in [rε , rε (vε )] except at y = rε (vε ), where it is zero. Thus G(y)
decreases strictly in the above interval, meaning
Fλ1 ,ε (rε (vε )vε ) < Fλ1 ,ε (rε vε ) ≤ a0 ,
(2.17)
0
since Fλ1 ,ε (rε vε ) ≤ lim inf n→+∞ Fλ1 ,ε (rε (vn )vn ) = a . Next we choose a positive
k, such that kvε ∈ W . Since (2.4) holds, we arrive at an obvious contradiction.
Thus rε = rε (vε ), and Fλ1 ,ε (rε (kvε )kvε ) = a0 , thus obtaining a conditional critical
point of Fbλ1 ,ε (·) subject to L(v) = 1, and, consequently, uε := rε (vε )vε is a critical
point of Fλ1 ,ε (·). Once more, we may assume vε ≥ 0, and so uε is a nontrivial
nonnegative solution of (2.9).
Next we study the behavior of the solutions uε = rε (vε )vε as ε → 0. Let ϕ1 > 0
be the eigenfunction of (1.3) corresponding to λ1 , with L(ϕ1 ) = 1. By (2.11),
R
b(x)|ϕ1 |q dx
s−q
,
(2.18)
rε (ϕ1 )
= RΩ
ε Ω d(x)|ϕ1 |s dx
which implies that rε (ϕ1 ) → +∞ as ε → 0. In view of (2.12) and (2.11)
Z
1 1
1 1
Fλ1 ,ε (rε (ϕ1 )ϕ1 ) = ε − rε (ϕ1 )s D(ϕ1 ) =
− rε (ϕ1 )q
b(x)|ϕ1 |q dx.
s q
s q
Ω
Since Fλ1 ,ε (rε (vε )vε ) ≤ Fλ1 ,ε (rε (ϕ1 )ϕ1 ), we conclude that
Fλ1 ,ε (uε ) = Fλ1 ,ε (rε (vε )vε ) → −∞
6
D. A. KANDILAKIS, M. MAGIROPOULOS
EJDE-2011/95
as ε → 0. By (2.12) we also get that rε (vε ) → +∞ as ε → 0. Let vb be a weak
accumulation point of vε ; that is, vb = w − limn→+∞ vεn where εn → 0 as n → +∞.
Since L(vεn ) = 1, necessarily
Z
Z
Z
0≤
|∇vεn |p dx +
a(x)|vεn |p dx − λ1
c(x)|vεn |p dσ(x) → 0.
Ω
Ω
∂Ω
Consequently, either vb = 0 or vb = γϕ1 for some
R γ 6= 0. sWe cannot
R have vb = 0, because then, since vεn ∈ W , we would get that Ω d(x)|b
v | dx = lim Ω d(x)|vεn |s dx =
1. Therefore, vb = γϕ1 and so kuεn k = rεn (vεn )kvεn k → +∞ as n → +∞.
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Dimitrios A. Kandilakis
Department of Sciences, Technical University of Crete, 73100 Chania, Greece
E-mail address: dkan@science.tuc.gr
Manolis Magiropoulos
Department of Sciences, Technological Educational Institute of Crete, 71500 Heraklion, Greece
E-mail address: mageir@staff.teicrete.gr