On positive solutions of p-Laplacian-type equations

0.28 MB·English

by  Yehuda Pinchover

On positive solutions of p-Laplacian-type
9 equations
0
0
2
n Yehuda Pinchover Kyril Tintarev
a
J Department of Mathematics Department of Mathematics

7
Technion - Israel Institute of Technology Uppsala University

] Haifa 32000, Israel SE-751 06 Uppsala, Sweden
P
A [email protected] [email protected]
.
h
t January 7, 2009
a
m
[

1 Dedicated to Vladimir Maz’ya on the occasion of his 70th birthday
v
7
Abstract
4
8 Let Ω be a domain in Rd, d ≥ 2, and 1 < p < ∞. Fix V ∈ L∞(Ω).
0 loc
Consider the functional Q and its Gaˆteaux derivative Q′ given by
.
1
0 1
9 Q(u) := (|∇u|p+V|u|p)dx, Q′(u) :=−∇·(|∇u|p−2∇u)+V|u|p−2u.
p
0 ZΩ
:
v Inthispaperwediscussseveralaspectsofrelationsbetweenfunctional-
Xi analytic properties of the functional Q and properties of positive so-
lutions of the equation Q′(u) = 0.
r
a
2000 Mathematics Subject Classification. Primary 35J60; Secondary
35J20, 35J70, 49R50.
Keywords. quasilinear elliptic operator, p-Laplacian, ground state,
positive solutions, comparison principle, minimal growth.
1 Introduction and Preliminaries
Propertiesofpositivesolutionsofquasilinear ellipticequations, andinpartic-
ular of equations with the p-Laplacian term in the principal part, have been
1
extensively studied over the recent decades, (see for example [3, 4, 28, 32]
and the references therein). Fix p ∈ (1,∞), a domain Ω ⊆ Rd and a real
valued potential V ∈ L∞(Ω). The p-Laplacian equation in Ω with potential
loc
V is the equation of the form
−∆ (u)+V|u|p−2u = 0 in Ω, (1.1)
p
where∆ (u) := ∇·(|∇u|p−2∇u)isthecelebratedp-Laplacian. Thisequation,
p
in the semistrong sense, is a critical point equation for the functional
1
Q(u) = Q (u) := (|∇u|p +V|u|p) dx u ∈ C∞(Ω). (1.2)
V p 0
ZΩ
So, we consider solutions of (1.1) in the following weak sense.
Definition 1.1. A function v ∈ W1,p(Ω) is a (weak) solution of the equation
loc
Q′(u) := −∆ (u)+V|u|p−2u = 0 in Ω, (1.3)
p
if for every ϕ ∈ C∞(Ω)
0
(|∇v|p−2∇v ·∇ϕ+V|v|p−2vϕ)dx = 0. (1.4)
ZΩ
We say that a real function v ∈ C1 (Ω) is a supersolution (resp. subsolution)
loc
of the equation (1.3) if for every nonnegative ϕ ∈ C∞(Ω)
0
(|∇v|p−2∇v ·∇ϕ+V|v|p−2vϕ)dx ≥ 0 (resp. ≤ 0). (1.5)
ZΩ
Next, we mention local properties of solutions of (1.3) that hold in any
smooth subdomain Ω′ ⋐ Ω (i.e., Ω′ is compact in Ω).
1. Smoothness and Harnack inequality. Weak solutions of (1.3) admit
H¨older continuous first derivatives, and nonnegative solutions of (1.3) satisfy
the Harnack inequality (see for example [14, 34, 35, 32, 38]).
2. Principal eigenvalue and eigenfunction. For any smooth subdomain
Ω′ ⋐ Ω consider the variational problem
(|∇u|p+V|u|p)dx
λ (Ω′) := inf Ω′ . (1.6)
1,p
u∈W01,p(Ω′) R Ω′|u|pdx
R
2
It is well-known that for such a subdomain, (1.6) admits (up to a multiplica-
tive constant) aunique minimizer ϕ [9,13]. Moreover, ϕ isa positive solution
of the quasilinear eigenvalue problem
Q′(ϕ) = λ (Ω′)|ϕ|p−2ϕ in Ω′,
1,p
(1.7)
(ϕ = 0 on ∂Ω′.
λ (Ω′) and ϕ are called, respectively, the principal eigenvalue and eigen-
1,p
function of the operator Q′ in Ω′.
3. Weak and strong maximum principles.
Theorem 1.2 ([13] (see also [3, 4])). Assume that Ω ⊂ Rd is a bounded
C1+α-domain, where 0 < α ≤ 1. Consider a functional Q of the form (1.2)
with V ∈ L∞(Ω). The following assertions are equivalent:
(i) Q′ satisfies the maximum principle: If u is a solution of the equation
Q′(u) = f ≥ 0 in Ω with some f ∈ L∞(Ω), and satisfies u ≥ 0 on ∂Ω,
then u is nonnegative in Ω.
(ii) Q′ satisfies the strong maximum principle: If u is a solution of the
equation Q′(u) = f (cid:9) 0 in Ω with some f ∈ L∞(Ω), and satisfies u ≥ 0
on ∂Ω, then u > 0 in Ω.
(iii) λ (Ω) > 0.
1,p
(iv) For some 0 (cid:8) f ∈ L∞(Ω) there exists a positive strict supersolution v
satisfying Q′(v) = f in Ω, and v = 0 on ∂Ω.
(iv’) There exists a positive strict supersolution v ∈ W1,p(Ω)∩L∞(Ω) satis-
fying Q′(v) = f (cid:9) 0 in Ω, such that v| ∈ C1+α(∂Ω) and f ∈ L∞(Ω).
∂Ω
(v) For each nonnegative f ∈ Cα(Ω) ∩ L∞(Ω) there exists a unique weak
nonnegative solution of the problem Q′(u) = f in Ω, and u = 0 on ∂Ω.
4. Weak comparison principle. We recall also the following weak com-
parison principle (or WCP for brevity).
Theorem 1.3 ([13]). Let Ω ⊂ Rd be a bounded domain of class C1,α, where
0 < α ≤ 1, and suppose that V ∈ L∞(Ω). Assume that λ (Ω) > 0 and let
1,p
3
u ∈ W1,p(Ω)∩L∞(Ω) satisfying Q′(u ) ∈ L∞(Ω), u | ∈ C1+α(∂Ω), where
i i i ∂Ω
i = 1,2. Suppose further that the following inequalities are satisfied
Q′(u ) ≤ Q′(u ) in Ω,
1 2
Q′(u ) ≥ 0 in Ω,
 2 (1.8)
u ≤ u on ∂Ω,
 1 2

 u ≥ 0 on ∂Ω.
2

Then 
u ≤ u in Ω.
1 2
5. Strong comparison principle.
Definition 1.4. we say that the strong comparison principle (or SCP for
brevity) holds true for the functional Q if the conditions of Theorem 1.3
implies that u < u in Ω unless u = u in Ω.
1 2 1 2
Remark 1.5. It is well known that the SCP holds true for p = 2 and for
p-harmonic functions. For sufficient conditions for the validity of the SCP
see [2, 5, 7, 16, 32, 37] and the references therein. In [5] M. Cuesta and
P. Tak´aˇc present a counterexample where the WCP holds true but the SCP
does not.
Throughout this paper we assume that
Q(u) ≥ 0 ∀u ∈ C∞(Ω). (1.9)
0
The following Allegretto-Piepenbrink-type theorem, links the existence of
positive solutions with the positivity of Q.
Theorem 1.6 ([28, Theorem 2.3]). Consider a functional Q of the form
(1.2). The following assertions are equivalent:
(i) The functional Q is nonnegative on C∞(Ω).
0
(ii) Equation (1.3) admits a global positive solution.
(iii) Equation (1.3) admits a global positive supersolution.
In this paper we survey further connections between functional-analytic
properties of the functional Q and properties of its positive solutions. In
particular, we review the following topics:
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• A representation of the nonnegative functional Q as an integral of a
nonnegative Lagrangiandensity andtheexistence of auseful equivalent
nonnegative Lagrangian density with a simplified form (Section 2).
• The equivalence of several weak coercivity properties of Q. The char-
acterization of the non-coercive case in terms of a Poincar´e-type in-
equality, in terms of the existence of a generalized ground state, and in
terms of the variational capacity of balls (Section 3).
• The identification of ground state as a global minimal solution (Section
4).
• AtheoremofLiouvilletypeconnectingthebehaviorofagroundstateof
onefunctionalwiththeexistenceofagroundstateofanotherfunctional
with a given ‘decaying’ subsolution (Section 5).
• A variational principle that characterizes solutions of minimal growth
at infinity (Section 6).
• The existence of solutions to the inhomogeneous equation Q′(u) = f
in the absence of the ground state (Section 7).
• The dependence of weak coercivity on the potential and the domain
(Section 8). In particular, in Theorem 8.6, we extend the result for
p = 2 proved in [30, Theorem 2.9].
• Properties verified only in the linear case (p = 2), in particular, the
definition of a natural functional space associated with the functional
Q (Section 9).
2 Positive Lagrangian representations
Let v ∈ C1 (Ω) be a positive solution (resp. subsolution) of (1.3). Using
loc
the positive Lagrangian representation from [3, 4, 6], we infer that for every
u ∈ C∞(Ω), u ≥ 0,
0
Q(u) = L(u,v)dx, resp. Q(u) ≤ L(u,v)dx, (2.1)
ZΩ ZΩ
5
where
1 up up−1
L(u,v) := |∇u|p+(p−1) |∇v|p −p ∇u·|∇v|p−2∇v . (2.2)
p vp vp−1
(cid:20) (cid:21)
It can be easily verified that L(u,v) ≥ 0 in Ω [3, 4].
Let now w := u/v, where v is a positive solution of (1.3) and u ∈ C∞(Ω),
0
u ≥ 0. Then (2.1) implies that
1
Q(vw) = |w∇v+v∇w|p−wp|∇v|p −pwp−1v|∇v|p−2∇v ·∇w dx.
p
ZΩ
(cid:2) (cid:3) (2.3)
Similarly, if v is a nonnegative subsolution of (1.3), then
1
Q(vw) ≤ |w∇v+v∇w|p−wp|∇v|p −pwp−1v|∇v|p−2∇v ·∇w dx.
p
ZΩ
(cid:2) (cid:3) (2.4)
Therefore, a nonnegative functional Q can be represented as the integral of
a nonnegative Lagrangian L. In spite of the nonnegativity of the in (2.1)
and (2.3), the expression (2.2) of L contains an indefinite term which poses
obviousdifficulties forextending thedomainofthefunctionaltomoregeneral
weakly differentiable functions. The next proposition shows that Q admits
a two-sided estimate by a simplified Lagrangian containing only nonnegative
terms. We call the functional associated with this simplified Lagrangian the
simplified energy.
Let f and g be two nonnegative functions. We denote f ≍ g if there
exists a positive constant C such that C−1g ≤ f ≤ Cg.
Proposition 2.1 ([26, Lemma 2.2]). Let v ∈ C1 (Ω) be a positive solution
loc
of (1.3). Then
Q(vw) ≍ v2|∇w|2(w|∇v|+v|∇w|)p−2 dx ∀w ∈ C1(Ω),w ≥ 0. (2.5)
0
ZΩ
In particular, for p ≥ 2, we have
Q(vw) ≍ vp|∇w|p +v2|∇v|p−2wp−2|∇w|2 dx ∀w ∈ C1(Ω),w ≥ 0.(2.6)
0
ZΩ
(cid:0) (cid:1)
If v is only a nonnegative subsolution of (1.3), then for 1 < p < ∞ we
have
Q(vw)≤C v2|∇w|2(w|∇v|+v|∇w|)p−2dx ∀w ∈ C1(Ω),w ≥ 0.
0
ZΩ∩{v>0}
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In particular, for p ≥ 2 we have
Q(vw) ≤ C vp|∇w|p +v2|∇v|p−2wp−2|∇w|2 dx ∀w ∈ C1(Ω),w ≥ 0.
0
ZΩ
(cid:0) (cid:1)
Remark 2.2. It is shown in [26] that for p > 2 neither of the terms in the
simplified energy (2.6) is dominated by the other, so that (2.6) cannot be
further simplified.
3 Coercivity and ground state
It is well known (see [21]) that for a nonnegative Schr¨odinger operator P we
have the following dichotomy: either there exists a strictly positive potential
W such that the Schr¨odinger operator P − W is nonnegative, or P admits
a unique (generalized) ground state. It turns out that this statement is also
true for nonnegative functionals of the form (1.2).
Definition 3.1. Let Q be a nonnegative functional on C∞(Ω) of the form
0
(1.2). We say that a sequence {u } ⊂ C∞(Ω) of nonnegative functions is a
k 0
null sequence of the functional Q in Ω, if there exists an open set B ⋐ Ω such
that |u |pdx = 1, and
B k
R
lim Q(u ) = lim (|∇u |p +V|u |p)dx = 0. (3.1)
k k k
k→∞ k→∞
ZΩ
We say that a positive function v ∈ C1 (Ω) is a ground state of the functional
loc
Q in Ω if v is an Lp (Ω) limit of a null sequence of Q. If Q ≥ 0, and Q admits
loc
a ground state in Ω, we say that Q is critical in Ω.
Remark 3.2. Therequirement that{u } ⊂ C∞(Ω), canclearlybeweakened
k 0
by assuming onlythat {u } ⊂ W1,p(Ω). Also, as it follows fromTheorem 3.4,
k 0
the requirement |u |pdx = 1 can be replaced by |u |pdx ≍ 1 or by
B k B k
u dx ≍ 1.
B k R R
R The following statements are based on rephrased statements of [28, The-
orem 1.6], [29, Theorem 4.3] and [36, Proposition 3.1] (cf. [21, 27] for the
case p = 2).
Theorem 3.3. Suppose that the functional Q is nonnegative on C∞(Ω).
0
7
1. Any ground state v is a positive solution of (1.3).
2. Q admits a ground state v if and only if (1.3) admits a unique positive
supersolution.
3. Q is critical in Ω if and only if Q admits a null sequence that converges
locally uniformly in Ω.
4. If Q admits a ground state v, then the followingPoincar´etype inequality
holds: There exists a positive continuous function W in Ω, such that
for every ψ ∈ C∞(Ω) satisfying ψvdx 6= 0 there exists a constant
0
C > 0 such that the following inequality holds:
R
p
Q(u)+C ψudx ≥ C−1 W (|∇u|p+|u|p) dx ∀u ∈ C∞(Ω).
0
(cid:12)ZΩ (cid:12) ZΩ
(cid:12) (cid:12) (3.2)
(cid:12) (cid:12)
(cid:12) (cid:12)
The following theorem slightly extends [28, Theorem 1.6] in the spirit of
[36, Proposition 3.1].
Theorem 3.4. Suppose that the functional Q is nonnegative on C∞(Ω). The
0
following statements are equivalent.
(a) Q does not admit a ground state in Ω.
(b) There exists a continuous function W > 0 in Ω such that
Q(u) ≥ W(x)|u(x)|pdx ∀u ∈ C∞(Ω). (3.3)
0
ZΩ
(c) There exists a continuous function W > 0 in Ω such that
Q(u) ≥ W(x)(|∇u(x)|p +|u(x)|p) dx ∀u ∈ C∞(Ω). (3.4)
0
ZΩ
(d) There exists an open set B ⋐ Ω and C > 0 such that
B
p
Q(u) ≥ C u(x)dx ∀u ∈ C∞(Ω). (3.5)
B 0
(cid:12)ZB (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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Suppose further that d > p. Then Q does not admit a ground state in Ω if
and only if there exists a continuous function W > 0 in Ω such that
p/p∗
Q(u) ≥ W(x)|u(x)|p∗dx ∀u ∈ C∞(Ω), (3.6)
0
(cid:18)ZΩ (cid:19)
where p∗ = pd/(d−p) is the critical Sobolev exponent.
Definition 3.5. A nonnegative functional Q on C∞(Ω) of the form (1.2)
0
which is not critical is said to be subcritical (or weakly coercive) in Ω.
Example 3.6. Consider the functional Q(u) := |∇u|pdx. It follows from
Rd
[20, Theorem 2] that if d ≤ p, then Q admits a ground state ϕ = constant in
R
Rd. On the other hand, if d > p, then
u(x) := 1+|x|p/(p−1) (p−d)/p, v(x) := constant
are two positive supe(cid:2)rsolutions of(cid:3)the equation −∆ u = 0 in Rd. Therefore,
p
Q is weakly coercive in Rd.
Example 3.7. Let d > 1, d 6= p, and Ω := Rd\{0} be the punctured space.
The following celebrated Hardy’s inequality holds true:
|u|p
Q (u) := |∇u|p −λ dx ≥ 0 u ∈ C∞(Ω), (3.7)
λ |x|p 0
ZΩ(cid:18) (cid:19)
whenever λ ≤ c∗ := |(p−d)/p|p. Clearly, if λ < c∗ , then Q (u) is weakly
p,d p,d λ
coercive. On the other hand, the proof of Theorem 1.3 in [31] shows that
Q with λ = c∗ admits a null sequence. It can be easily checked that the
λ p,d
function v(r) := |r|(p−d)/p is a positive solution of the corresponding radial
equation:
d−1 |v|p−2v
−|v′|p−2 (p−1)v′′ + v′ −c∗ = 0 r ∈ (0,∞).
r p,d rp
(cid:20) (cid:21)
Therefore, ϕ(x) := |x|(p−d)/p is the ground state of the equation
|u|p−2u
−∆ u−c∗ = 0 in Ω. (3.8)
p p,d |x|p
Note that ϕ 6∈ W1,p(Rd) for p 6= d. In particular, ϕ is not a positive superso-
loc
lution of the equation ∆ u = 0 in Rd.
p
9
In [39, 40] Troyanov has established a relationship between the p-capacity
ofclosedballsinaRiemannianmanifoldMandthep-parabolicityofMwith
respect the p-Laplacian. We extend his definition and result to our case.
Definition 3.8. Suppose that the functional Q is nonnegative on C∞(Ω).
0
Let K ⋐ Ω be a compact set. The Q-capacity of K in Ω is defined by
Cap (K,Ω) := inf{Q(u) | u ∈ C∞(Ω), u ≥ 1 on K}.
Q 0
Corollary 3.9. Suppose that the functional Q is nonnegative on C∞(Ω).
0
Then Q is critical in Ω if and only if the Q-capacity of each closed ball in Ω
is zero.
4 Ground states and minimal growth at in-
finity
Definition 4.1. Let K be a compact set in Ω. A positive solution u of
0
the equation Q′(u) = 0 in Ω\K is said to be a positive solution of minimal
0
growth in a neighborhood of infinity in Ω (oru ∈ M for brevity) if for any
Ω,K0
compact set K in Ω, with a smooth boundary, such that K ⋐ int(K), and
0
any positive supersolution v ∈ C((Ω\ K)∪∂K) of the equation Q′(u) = 0
in Ω\K, the inequality u ≤ v on ∂K implies that u ≤ v in Ω\K.
A (global) positive solution u of the equation Q′(u) = 0 in Ω, which has
minimal growth in a neighborhood of infinity in Ω (i.e. u ∈ M ) is called
Ω,∅
a global minimal solution of the equation Q′(u) = 0 in Ω.
Theorem 4.2 ([29, Theorem 5.1], cf. [28]). Suppose that 1 < p < ∞, and Q
is nonnegative on C∞(Ω). Then for any x ∈ Ω the equation Q′(u) = 0 has
0 0
a positive solution u ∈ M .
Ω,{x0}
We have the following connection between the existence of a global min-
imal solutions and weak coercivity.
Theorem 4.3 ([29, Theorem 5.2], cf. [28]). Let 1 < p < ∞, and assume
that Q is nonnegative on C∞(Ω). Then Q is subcritical in Ω if and only if
0
the equation Q′(u) = 0 in Ω does not admit a global minimal solution of the
equation Q′(u) = 0 in Ω. In particular, u is ground state of the equation
Q′(u) = 0 in Ω if and only u is a global minimal solution of this equation.
10