On Singular p-Laplacian Problems Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, USA
[email protected] Elves A. B. Silva Departamento de Matem´atica Universidade de Bras´ılia 70910-900, DF, Brazil
[email protected] Abstract In this work we combine perturbation arguments and variational methods to prove the existence and uniqueness results for singular p-Laplacian problems. MSC2000: 34B16 Key Words and Phrases: singular p-Laplacian problems, existence and uniqueness
1
1
Introduction
Consider the boundary value problem − ∆p u = f (x, u) in Ω u>0 in Ω u=0 on ∂Ω
(1.1)
where Ω is a bounded domain in Rn , n ≥ 1, ∆p u = div (|∇u|p−2 ∇u) is the p-Laplacian of u, 1 < p < ∞, and f is a Carath´eodory function on Ω×(0, ∞) satisfying (f1 ) given 0 < t1 ≤ t2 < ∞, there are h1 ∈ L1loc (Ω) and h2 ∈ L1 (Ω) such that −h1 (x) ≤ f (x, t) ≤ h2 (x) for all 0 < t1 ≤ t ≤ t2 , a.e. in Ω, (f2 ) there are a nontrivial function a ≥ 0 in L1 (Ω) and t0 > 0 such that f (x, t) ≥ a(x) for all 0 < t ≤ t0 , a.e. in Ω, (f3 ) there is a nontrivial function α ≥ 0 such that lim inf t→∞
λ1 tp−1 − f (x, t) ≥ α(x), unif. a.e. in Ω, tp−2
where λ1 > 0 is the first eigenvalue of the operator −∆p in W01,p (Ω). Note that, under the conditions (f1 ) and (f2 ), the function f may be singular at t = 0 and it may change sign. The condition (f3 ) allows the Problem (1.1) to be resonant at the first eigenvalue λ1 . Note also that the conditions (f1 ) − (f3 ) do not imply any growth restriction from below on f with respect to the variable t. In a pioneering paper Crandall, Rabinowitz, and Tartar [6] considered the semilinear case p = 2, with a more general second order uniformly elliptic operator satisfying a maximum principle in place of the Laplacian and under the stronger assumptions ∂Ω ∈ C 3 , f ∈ C(Ω × (0, ∞)), lim f (x, t) = ∞ uniformly in Ω,
t→0+
sup f < ∞, [1,∞)×Ω
2
2, q and obtained a generalized solution in Wloc (Ω) ∩ C0 (Ω) for some q > n, satisfying (1.1) almost everywhere. They further showed that this solution is a classical C 2 (Ω) solution if f ∈ C 1 (Ω × (0, ∞)) and is unique if in addition f is nonincreasing in t. Since then singular equations of the form
−∆ u = a(x) u−γ + µ g(x, u)
(1.2)
where a ≥ 0 is a nontrivial function in L2 (Ω), γ > 0 is a constant, µ ≥ 0 is a parameter, and g ≥ 0 is a Carath´eodory function on Ω × [0, ∞) have been studied extensively (see, e.g., [3 - 5, 7, 10, 11, 13, 17, 19, 21] and their references). Similar equations for the p-Laplacian and small γ were studied by Agarwal, Perera, and O’Regan [1], Perera and Silva [14, 15], and Perera and Zhang [16]. In the present paper we consider the problem − ∆p u = f (x, u) + µ g(x, u) in Ω u>0 in Ω (1.3) u=0 on ∂Ω where f satisfies (f1 ) - (f3 ), µ ≥ 0 is a parameter, and g is a Carath´eodory function on Ω × [0, ∞) satisfying (g1 ) given t1 > 0, there are h3 ∈ L1loc (Ω) and h4 ∈ Lq1 (Ω), q1 = 1 if p > n, and q1 > n/p if p ≤ n, such that −h3 (x) ≤ g(x, t) ≤ h4 (x) for all 0 ≤ t ≤ t1 , a.e. in Ω, (g2 ) there is c1 > 0 such that g(x, t) ≥ −c1 a(x) for all 0 ≤ t ≤ t0 , a.e. in Ω, Note that conditions (g1 ) − (g2 ) does not impose any grwoth restriction on g with respect to the variable t. In view of conditions (f1 ), (f2 ) and (g1 ), we look for a solution of Problem (1.3) that solves the quasilinear equation in the sense of distributions and satisfies the boundary condition in a more general sense. More specifically, we define: 3
1, p Definition 1.1. By a solution of (1.3) we mean a function u ∈ Wloc (Ω) such that
(i) ess inf u > 0 for all Ω0 ⊂⊂ Ω, 0 Ω
Z (ii)
|∇u|p−2 ∇u · ∇ϕ − (f (x, u) + µ g(x, u)) ϕ = 0 for all ϕ ∈ C0∞ (Ω),
Ω
(iii) (u − ε)+ ∈ W01, p (Ω) for all ε > 0. Actually we will show that u = 0 on ∂Ω whenever p > n or h2 and h4 are in L∞ (Ω) and ∂Ω is regular (see [20]) in the sense that lim inf + ρ→0
|Kx (ρ) \ Ω| > 0 for all x ∈ ∂Ω, ρn
where Kx (ρ) denotes the cube with center x and sides of length ρ parallel to the coordinate axes. Now we may state: Theorem 1.2. If (f1 ) - (f3 ), (g1 ), and (g2 ) hold, then there is a µ0 > 0 1, p such that the problem (1.3) has a solution u ∈ Wloc (Ω) ∩ L∞ (Ω) for all 0 ≤ µ < µ0 . If p > n or ∂Ω is regular and (f1 ), (g1 ) hold with h2 , h4 ∈ L∞ (Ω), respectively, then u = 0 on ∂Ω. Supposing the following stronger versions of (f1 ) and (g1 ), 2 (fˆ1 ) given 0 < t1 ≤ t2 < ∞, there are h1 ∈ Lqloc (Ω) h2 ∈ Lq2 (Ω) and q2 > pn(p − 1)−1 , such that
−h1 (x) ≤ f (x, t) ≤ h2 (x) for all 0 < t1 ≤ t ≤ t2 , a.e. in Ω, 3 (ˆ g1 ) given t1 > 0, there are h3 ∈ Lqloc (Ω) and h4 ∈ Lq3 (Ω), q3 > pn(p − 1)−1 , such that
−h3 (x) ≤ g(x, t) ≤ h4 (x) for all 0 ≤ t ≤ t1 , a.e. in Ω, we may combine Theorem 1.2 and the local regularity theory results of DiBenedetto [8] to obtain
4
Theorem 1.3. If (fˆ1 ), (f2 ), (f3 ), (ˆ g1 ), and (g2 ) hold, then there is a µ0 > 0 1, α such that the problem (1.3) has a solution u ∈ Cloc (Ω) ∩ L∞ (Ω) for all 0 ≤ µ < µ0 . If p > n or ∂Ω is regular and (fˆ1 ), (ˆ g1 ) hold with h2 , h4 ∈ L∞ (Ω), respectively, then u ∈ C0 (Ω). Finally, we present a uniqueness result for solutions of Problem (1.1). Theorem 1.4. If (f1 ) holds and f is nonincreasing in t, then the problem 1, p (1.1) has at most one solution u ∈ Wloc (Ω) satisfying u = 0 on ∂Ω. Example 1.5. There is a µ0 > 0 such that the problem − ∆p u = e1/u + µ eu in Ω u>0 in Ω u=0 on ∂Ω
(1.4)
1, α has a solution u ∈ Cloc (Ω) ∩ L∞ (Ω) for all µ < µ0 . If p > n or ∂Ω is regular, then u ∈ C0 (Ω) and is unique when µ ≤ 0.
2
Preliminaries
In order to solve Problem (1.3), we approximate it by the sequence of problems ( − ∆p u = fj (x, u) + µ gj (x, u) in Ω (2.1) u=0 on ∂Ω where fj (x, t) = min{max{f (x, (t − εj )+ + εj ), −ε−1 j }, gj (x, t) = max{g(x, t), −c1 ε−1 j },
(2.2)
with c1 > 0 given by (g2 ) and (εj ) a sequence on (0, ∞) such that εj → 0+ as j → ∞. Note that, fj (x, t) and gj (x, t) are Carath´eodory function on Ω × R which satisfy fj (x, t) → f (x, t) and gj (x, t) → g(x, t), as j → ∞, for every (x, t) ∈ Ω × (0, ∞).
5
Lemma 2.1. If (f1 ) and (g1 ) hold, µ ≥ 0, and (uj ) ⊂ W01, p (Ω) is a sequence of weak solutions of problems (2.1) satisfying δΩ0 := inf ess inf uj > 0 for all Ω0 ⊂⊂ Ω, 0
(2.3)
M := sup kuj k∞ < ∞,
(2.4)
j
Ω
j
then a subsequence of (uj ) converges almost everywhere to a solution 1,p u ∈ Wloc (Ω) ∩ L∞ (Ω) of (1.3). If p > n or ∂Ω is regular and (f1 ), (g1 ) hold with h2 , h4 ∈ L∞ (Ω), respectively, then u = 0 on ∂Ω. Proof. Let (Ωk ) be [ a sequence of open subsets of Ω such that Ωk ⊂⊂ Ωk+1 for each k and Ωk = Ω. Set δk = δΩk , δΩk given by (2.3). Taking k
ϕ = (uj − δ1 )+ in Z |∇uj |p−2 ∇uj · ∇ϕ − (fj (x, uj ) + µ gj (x, uj )) ϕ = 0
(2.5)
Ω
gives Z
p
Z
Z
|∇uj | =
|∇uj | ≤ Ω1
p
uj >δ1
(fj (x, uj ) + µ g(x, uj )) (uj − δ1 ). (2.6) uj >δ1
The last integral is bounded by (2.2), (2.4), (f1 ), and (g1 ), so (uj ) is bounded in W 1, p (Ω1 ) and, consequently, it has a subsequence (ujk1 ) which converges weakly in W 1, p (Ω1 ), strongly in Lp (Ω1 ), and almost everywhere in Ω1 . Denote by uΩ1 the corresponding limit in W 1, p (Ω1 ). Arguing by induction, for each k we obtain a subsequence (ujlk ) of (uj ) and uΩk ∈ W 1, p (Ωk ) such that (ujlk ) converges to uΩk weakly in W 1, p (Ωk ), strongly in Lp (Ωk ), and almost everywhere in Ωk . Moreover we may assume that (uj k+1 ) is a subsequence of (ujlk ), for every k, and that jkk → ∞ as l k → ∞. By construction uΩk+1 |Ωk = uΩk , so setting u = uΩ1 and u = uΩk+1 on 1, p (Ω) ∩ L∞ (Ω), to Ωk+1 \ Ωk for each k, we get a well-defined function u ∈ Wloc 1, p which the diagonal subsequence (ujk ) := (ujkk ) converges weakly in Wloc (Ωk ), p strongly in Lloc (Ωk ), and almost everywhere in Ω. We claim that, actually, 1, p the diagonal sequence (ujk ) converges strongly to u in Wloc (Ωk ). 1, p Assuming the claim, we may verify that the function u ∈ Wloc (Ω)∩L∞ (Ω) is a solution of (1.3), i.e. u satisfies the conditions (i)-(iii) given in Definition 6
1.1. First note that u satisfies the condition (i) since (uj ) satisfies (2.3) and (ujk ) converges to u almost everywhere in Ω. Next, given ϕ ∈ C0∞ (Ω), fix k1 ≥ 1 such that supp ϕ ⊂ Ωk1 . Hence the relation (2.4) with jk in place of j reduces to Z |∇ujk |p−2 ∇ujk · ∇ϕ − (fjk (x, ujk ) + µ gjk (x, ujk )) ϕ = 0. (2.7) Ωk1
As |(fjk (x, ujk ) + µ gjk (x, ujk )) ϕ is bounded by a L1 function in Ω, from (f1 ), (g1 ), (2.2)-(2.4), and the above claim; we may pass to the limit in (2.7), obtaining Z |∇u|p−2 ∇u · ∇ϕ − (f (x, u) + µ g(x, u)) ϕ = 0 (2.8) Ω
since suppϕ ⊂ Ωk1 . Therefore the condition (ii) holds. Now, given ε > 0, we may argue as in (2.5) to show that ((ujk − ε)+ ) is bounded in W01, p (Ω), and hence it has a subsequence which converges to some v ∈ W01, p (Ω) weakly in W01, p (Ω) and almost everywhere in Ω. Then v = (u − ε)+ since (ujk − ε)+ → (u − ε)+ almost everywhere in Ω. This implies that the condition (iii) holds. Now we prove the claim. Let Ω0 ⊂⊂ Ω. We must show that (ujk |Ω0 ) converges strongly to u|Ω0 in W 1, p (Ω0 ). take η ∈ C0∞ (Ω, [0, 1]) be such that η = 1 on Ω0 , and consider k1 ≥ 1 such that supp η ⊂ Ωk1 . For every k, m ≥ 1, we have Z |∇ujk |p−2 ∇ujk − |∇ujm |p−2 ∇ujm · ∇(ujk − ujm ) ≤ Ω0
Z
|∇ujk |p−2 ∇ujk − |∇ujm |p−2 ∇ujm · ∇(η(ujk − ujm ))−
(2.9)
Ω
Z
|∇ujk |p−2 ∇ujk − |∇ujm |p−2 ∇ujm · ∇η (ujk − ujm ).
Ωk1
Using the fact that (ujk ) is bounded in W 1, p (Ωk1 ) and converges strongly in Lp (Ωk1 ), we obtain Z |∇ujk |p−2 ∇ujk − |∇ujm |p−2 ∇ujm · ∇η (ujk − ujm ) → 0, (2.10) Ωk1
as k, m → ∞. On the other hand, taking ϕ = η(ujk − ujm ) and either j = jk or j = jm in (2.5), gives us Z Z p−2 | |∇ujl | ∇ujl ·∇(η(ujk −ujm ))| ≤ |(fjl (x, ujl )+µ gjl (x, ujl ))(ujk −ujm )|, Ω
Ωk1
7
for l = k, m. Arguing as in the verification of (2.8), we pass to the limit in the above inequality to obtain Z |∇ujl |p−2 ∇ujl · ∇(η(ujk − ujm )) → 0, as k, m → ∞, (2.11) Ω
for l = k, m. The relations (2.9)-(2.11) and a standard argument (see e. g. [14] and the references there in) enable us to show that Z |∇ujk − ∇ujm |p → 0, as k, m → ∞. Ω0
This implies that (ujk ) is a Cauchy sequence in W 1, p (Ω0 ) since we already know that (ujk ) converges strongly in Lp (Ω0 ). The claim is proved. Our final task is to verify that u = 0 on ∂Ω if p > n or ∂Ω is regular and (f1 ), (g1 ) hold with h2 , h4 ∈ L∞ (Ω). Given 0 < ε < M := kuk∞ , we find hε > 0 ∈ Lq (Ω), q = 1 if p > n and q = ∞ if p ≤ n, such that f (x, t) + µg(x, t) ≤ hε (x), for every 0 < ε ≤ t ≤ M, a.e. in Ω. (2.12) 1,α Now let ϕε > 0 in W01, p (Ω) ∩ Cloc (Ω) be the unique solution of ( − ∆p ϕε = hε in Ω
ϕε = 0
(2.13)
on ∂Ω.
Then ϕε ∈ C0 (Ω) by the Sobolev imbedding theorem if p > n and by Trudinger [20] if ∂Ω is regular. Taking j = jk and ϕ = (ujk − ε − ϕε )+ in (2.5), and considering (2.2), (2.4) and (2.12), we obtain Z Z p−2 + |∇ujk | ∇ujk · ∇(ujk − ε − ϕε ) ≤ hε (x)(ujk − ε − ϕε )+ . Ω
Ω
Therefore, since ϕε is a solution of (2.12), Z |∇ujk |p−2 ∇ujk − |∇(ε + ϕε )|p−2 ∇(ε + ϕε ) · ∇(ujk − ε − ϕε ) ≤ 0, ujk >ε+ϕε
and, consequently, ujk ≤ ε+ϕε . Thus 0 < u ≤ ε+ϕε almost everywhere in Ω. ¯ there is a neighborhood U of ∂Ω such that 0 < u(x) < 2ε Since ϕε ∈ C0 (Ω), for almost everywhere x ∈ U ∩ Ω. The fact that ε > 0 can be chosen arbitrarily small implies that u = 0 on ∂Ω. 8
Since a ≥ 0 is nontrivial, the function a0 (x) = min{a(x), t−1 0 } ≥ 0 is also ∞ nontrivial, furthermore, it belongs to L (Ω). Consequently the problem ( −∆p u0 = a0 (x) in Ω (2.14) u0 = 0 on ∂Ω has a unique solution u0 > 0 in W01, p (Ω) ∩ L∞ (Ω). Fixing 0 < λ0 < 1 so 1
small that u := λ0p−1 u0 ≤ t0 almost everywhere in Ω, for all j so large that εj ≤ t0 , we may invoke (f2 ), (g2 ) and (2.2) to obtain −∆p u − fj (x, u) − µ gj (x, u) ≤ −((1 − λ0 ) − c1 µ) a0 (x). Hence there is µ1 > 0 so that u is a subsolution of the problems (2.1) whenever 0 ≤ µ < µ1 . Now consider the problem ( − ∆p u = fj (x, u) + h4 (x)η(u) in Ω (2.15) u=0 on ∂Ω where h4 is given by (g1 ), and η : Ω → R is a continuous function satisfying (η1 ) 0 < η(t) ≤ 1 for all t ∈ R, and η(t) → 0, as t → ∞. tp−2 Here, we note that, without loss of generality, we suppose that h4 ≥ 1 in Ω. Lemma 2.2. If (f1 )-(f3 ), (g1 ), and (g2 ) hold and the problems (2.15) have a sequence of weak solutions (uj ) ⊂ W01, p (Ω) such that sup k(uj − ε)+ k < ∞
for some ε > 0,
(2.16)
j
then there is a µ0 > 0 such that the problem (1.3) has a solution u ∈ 1, p (Ω) ∩ L∞ (Ω) for all 0 ≤ µ < µ0 . If p > n or ∂Ω is regular and (f1 ), Wloc (g1 ) hold with h2 , h4 ∈ L∞ (Ω), respectively, then u = 0 on ∂Ω.
9
Proof. For all j so large that εj ≤ t0 , fj (x, t) ≥ a(x) ≥ ε0 a(x), for all − ∞ < t ≤ t0 , a.e. in Ω, by (f2 ). Hence uj ≥ u. We claim that (uj ) is bounded in L∞ (Ω). Effectively, if p > n, this assertion is a consequence of (2.16) and the Sobolev imbedding theorem. On the other hand, if p ≤ n, we invoke (f3 ), (g1 ), (η1 ), and (2.2) to find b ≥ ε > 0 and a1 ∈ Lr (Ω), with r > n/p, such that |fj (x, t) + h4 (x)η(t)| ≤ a1 (x)(1 + tp−1 ), for all t ≥ b, a.e. in Ω. Hence, by (2.16) and Lemma A.1 of the appendix, (uj ) is bounded in L∞ (Ω). The claim is proved. In view of the above claim, (g1 ), (2.2), and (η1 ), we may find 0 < µ0 < µ1 such that, for 0 ≤ µ < µ0 , −∆p uj − fj (x, uj ) − µ gj (x, uj ) ≥ (η(uj ) − µ) h4 (x) ≥ 0, so uj is a supersolution of (2.1). Thus (2.1) has a weak solution uj in the order interval [u, uj ] by a standard argument (see e.g. [16]), and the conclusion follows from Lemma 2.1.
3
Proofs
Proof of Theorem 1.2. We apply Lemma 2.2. Weak solutions of (2.15) are the critical points of the C 1 functional Z 1 Φj (u) = |∇u|p − Fˆj (x, u), for all u ∈ W01, p (Ω), Ω p Z t ˆ where Fj (x, t) = fˆj (x, s) ds, fˆj (x, t) = fj (x, t) + h4 (x)η(t). By (f1 ), (g1 ), 0
(2.2), (η1 ), and the Sobolev imbedding theorem, 1 Φj (u) ≥ kukp − Cj kuk p for some Cj > 0, so Φj is bounded from below and coercive, and hence it has a global minimizer uj since it is weak lower semicontinuity. 10
Now we fix 0 < ε < t0 such that Z Z p−1 α(x)ϕp−1 0 < λ1 ε ϕ1 (x) < 1 (x) dx,
(3.1)
Ω
Ω
where α is given by (f3 ) and ϕ1 > 0 is the first normalized eigenvalue of −∆p on W 1,p (Ω). Note that, replacing α by min{α, 1} if necessary, we may always suppose that α ∈ L∞ (Ω). In the following our objective is to prove that (2.16) holds for the value of ε given above. Arguing by contradiction, and taking a subsequence if necessary, we suppose k(uj − ε)+ k → ∞, as j → ∞.
(3.2)
Taking γ > 0, to be chosen posteriorly, by (f3 ) there is t∞ > t0 such that f (x, t) ≤ −(α(x) − γ)tp−2 + λ1 tp−1 , for all t ≥ t∞ , a.e. in Ω. Hence, by (2.2) and (η1 ), and taking t∞ larger if necessary, for every j ≥ 1, we have fˆj (x, t) ≤ −(α(x) − γh5 (x))tp−2 + λ1 tp−1 , for all t ≥ t∞ , a.e. in Ω., (3.3) where h5 := h4 + 1 ∈ Lq (Ω), q = 1 for p > n, and q > n/p for p ≤ n. Next, we invoke (f1 ), (g1 ), (2.2), and (η1 ) to find h = h(ε, γ) ∈ L1 (Ω) such that fˆj (x, t) ≤ h(x), for all ε ≤ t ≤ t∞ , a.e. in Ω.
(3.4)
Considering (3.3)-(3.4) and taking ϕ = vj := (uj − ε)+ in Z |∇uj |p−2 ∇uj · ∇ϕ − (fˆj (x, uj ) + 1) ϕ = 0 Ω
gives Z
p
(vj + ε)p−1 vj
kvj k ≤ λ1 uj >t∞
Z −
(3.5)
(α(x) − γh5 (x))(vj + ε)p−2 vj + C,
uj >t∞
where C denotes a generic positive constant that depends on ε and γ, but not on j. 11
Now, defining zj := kvj k−1 vj , for j ≥ 1, we may suppose there is z ∈ W01, p (Ω) such that kzk ≤ 1 and 1, p zj * z weakly in W0 (Ω), zj → z strongly in Lp (Ω), (3.6) zj (x) → z(x) a.e in Ω, |zj | ≤ ψq (x) ∈ Lq (Ω) a.e in Ω, where 1 ≤ q ≤ ∞ if p > n, 1 ≤ q < ∞ if p = n, 1 ≤ q < p∗ = pn/(n − p) if p < n. We claim that actually z = ϕ1 . Indeed, from (3.5) Z (vj + ε)p−1 zj 1 ≤ λ1 kvj kp−1 uj >t∞ Z h5 (x))(vj + ε)p−2 zj C . +γ + p−1 kvj kp kvj k uj >t∞ Hence, from (3.2), (3.6), and the variational characterization of λ1 λ1 kzkpp ≤ kzkp ≤ 1 ≤ λ1 kzkpp . Consequently z = ϕ1 since z ≥ 0 almost everywhere in Ω. The claim is proved. Invoking (3.5) and the variational characterization of λ1 one more time, we find a constant Cˆ > 0, not depending on j, such that Z (vj + ε)p−1 − vjp−1 )vj (vj + ε)p−2 zj ≤ λ1 α(x) kvj kp−1 kvj kp−1 Ω Ω Z h5 (x))(vj + ε)p−2 vj Cˆ +γ + . kvj kp−1 kvj kp−1 Ω
Z
Therefore, using (3.6), the above claim, and passing to the limit in the above inequality, Z Z Z p−1 p−1 0< α(x)ϕ1 (x) ≤ λ1 ε ϕ1 (x) + γ h5 (x)ϕp−1 (3.7) 1 (x). Ω
Ω
Ω
12
Hence, if γ > 0 is sufficiently small, (3.1) and (3.7) imply that Z Z p−1 0< α(x)ϕ1 (x) < α(x)ϕp−1 1 (x). Ω
Ω
This shows that k(uj − ε)+ k is bounded. Proof of Theorem 1.3. Consider µ0 > 0 given by Theorem 1.3. For every 1, p 0 ≤ µ < µ0 , the Problem (1.3) has a solution u ∈ Wloc (Ω) ∩ L∞ (Ω). Then q f (·, u) + µ g(·, u) ∈ Lloc (Ω), q > np/(p − 1), by (i) of Definition 1.1, (fˆ1 ), and 1, α (ˆ g1 ), and hence u ∈ Cloc (Ω) by the local regularity results of DiBenedetto [8]. ¯ if p > n or ∂Ω is regular The last part of Theorem 1.3 implies that u ∈ C0 (Ω) and (fˆ1 ), (ˆ g1 ) hold with h2 , h4 ∈ L∞ (Ω). 1, p Proof of Theorem 1.4. Let u1 and u2 be solutions of (1.1) in Wloc (Ω) such that u = 0 on ∂Ω. Then, given ε > 0, by compactness we find an open neighborhood U of ∂Ω such that
0 ≤ u1 < ε, i = 1, 2, a.e. in Ω, 1, p so ϕ = (u1 −u2 −ε)+ ∈ Wloc (Ω) has compact support in Ω. Taking u = u1 , u2 with this ϕ in (ii) of Definition 1.1 and subtracting gives Z |∇u1 |p−2 ∇u1 − |∇u2 |p−2 ∇u2 · ∇(u1 − u2 ) u1 >u2 +ε Z = (f (x, u1 ) − f (x, u2 )) (u1 − u2 − ε) ≤ 0 (3.8) u1 >u2 +ε
since f is nonincreasing in t, which implies that u1 ≤ u2 + ε. Since ε is arbitrary, u1 ≤ u2 , and the reverse inequality follows similarly.
Appendix Consider the boundary value problem − ∆p u = h(x, u) in Ω, u>0 in Ω, u=0 on ∂Ω,
(A.1)
where Ω is a bounded domain in Rn , n > 1, 1 < p ≤ n, and h is a Carath´eodory function on Ω × (0, ∞) satisfying
13
(h1 ) there are a ∈ Lr (Ω), r ≥ n/p (r > 1 if n = p) and b > 0 such that h(x, t) ≤ a(x)(1 + tp−1 ), for all b ≤ t < ∞, a.e. in Ω. Our main result in this appendix provides a L∞ bound for weak solutions of problem (A.1) in function of its norm on Lp (Ω). Lemma A.1. If (h1 ) holds and u ∈ W01, p (Ω) is a weak solution of problem (A.1), then u ∈ L∞ (Ω). Furthermore kuk∞ ≤ M < ∞, with the constant M depending only on b, Ω, a, and kukp We observe that the proof of Lemma A.1 will be established through the verification of two main steps. First we show that u ∈ Lq (Ω) for every q ∈ [1, ∞). Moreover kukq ≤ M1 ,
(A.2)
with the constant M1 > 0, depending only on b, Ω, q, kakr , and kukp , such that Our proof of (A.2) is based on a estimate due to Brezis and Kato [2]. Here we adapt the argument presented by Struwe [18] for the Laplacian operator (see also Guedda and Veron [9] where a corresponding argument for the p-Lapacian operator is employed). In our second and final step, we obtain a L∞ bound for the weak solutions of (A.1) by combining (A.2) and a L∞ estimate for functions in W01,p (Ω) provided by the following lemma (see Ladyzhenskaya and Uraltseva [12]). Lemma A.2. If u ∈ W01, p (Ω), p ≤ n and there is c0 ≥ 0 such that for c ≥ c0 , Z p |∇u|p ≤ γcα |Ac |(1− n +ε) , Ac
where Ac = {x ∈ Ω| u(x) > b}, ε > 0, 0 ≤ α ≤ ε + p, then u+ = max{u, 0} ∈ L∞ (Ω). Furthermore ku+ k∞ ≤ M < ∞, with the constant M depending only on γ, α, ε, c0 , Ω and kukL1 (Ac0 ) . Proof of Lemma A.1. We start by verifying the estimate (A.2). We note that in order to prove (A.2) it suffices to show that given q ∈ [1, ∞), then (u−b)+ ∈ Lq (Ω) and k(u − b)+ kq ≤ M2 < ∞,
(A.3)
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with the constant M2 depending only on b, Ω, q, a, and k(u − b)+ kp . Taking v = (u − b)+ and L > b, we define w ∈ W01,p (Ω) by w(x) = v(x) min{v ps (x), Lp }, for every x ∈ Ω.
(A.4)
Using the above definition, we have ∇w(x) = ∇v(x) min{v ps (x), Lp } + z(x), for a.e. x ∈ Ω,
(A.5)
where (
psv ps ∇v(x) if 0 ≤ v(x) < L,
z(x) =
(A.6) if v(x) ≥ L.
0
From (A.4)-(A.6), (h1 ) and the fact that u is a weak solution of (A.1), we obtain Z Z p ps p |∇v| min{v , L } + ps v ps |∇v|p ≤ Ω
0≤v 0, we find c = c(a) > 0 such that !n Z n
2a1
a(x) p
p
< ε.
{a(x)≥c}
Hence, invoking (A.9), H¨older’s inequality, the Sobolev imbedding W01, p (Ω) ,→ ∗ Lp (Ω), and taking ε > 0 sufficiently small, we obtain Z p∗ p p(s+1) s p∗ |(v min{v , L})| ≤ a3 (1 + kvkp(s+1) ), (A.10) Ω
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where a3 = a3 (s, b, Ω, a) (note that a similar argument provides A.10 when p = n). ∗ Then, letting L → ∞, we conclude that v s+1 ∈ Lp (Ω) whenever v ∈ Lp(s+1) (Ω). Moreover, p(s+1)
kv (s+1) kpp∗ ≤ a3 (1 + kvkp(s+1) ).
(A.11)
Finally, given q ∈ [1, ∞), we take s0 = 0 and si +1 = n(si−1 +1)/(n−p), 1 ≤ i ≤ j, such that sj > q. Applying (A.11) j times, we conclude that (A.3) must hold with M2 depending only on b, Ω, q a, and kvkp . This completes the first part of the proof of Lemma A.1. Now we shall establish the L∞ bound for the weak solutions of (A.1). As above we take v = (u − b)+ ∈ W01,p (Ω). Moreover, we set c0 = 1 and we we denote by Ac the set {x ∈ Ω; v(x) > c} for every c ≥ c0 . Invoking the fact that u is a weak solution of (A.1), (h1 ) and our choice of c0 , we obtain Z Z p |∇v| = |∇u|p−2 ∇u.∇((v − c)+ ) Ac
Ω
Z ≤
a(x)(1 + u Ac
p−1
(A.12)
Z )u ≤ 2
p
a(x)u . Ac
Now, considering q1 > 1 such that q1 < n(r − 1)[r(n − p)]−1 , from (A.12) and H¨older’s inequality, we get Z r−1 (A.13) |∇v|p ≤ 2kakr kukpq |Ac | rq1 . Ac
where q = rpq1 [(r−1)(q1 −1)]−1 > 1. Setting ε = [(r−1)n−(n−p)rq1 ](rnq1 )−1 > 0 and α = 0, we complete the proof of Lemma A.1 by invoking (A.2) and Lemma A.2. Remark A.3. If (h1 ) is satisfied with r > n/p, then Lemma A.1 holds with the constant M depending only on b, Ω, kakr , and kukp . Considering (h2 ) there are constants c, b1 > 0 and p < s < ∞ (s ≤ p∗ if p < n) such that h(x, t) ≤ c(1 + ts−1 ), t ∈ [b1 , ∞), As a direct consequence of Lemma A.1 and the above Remark, we have Lemma A.4. If (h2 ) holds and u ∈ W01, p (Ω) is a weak solution of problem (A.1), then u ∈ L∞ (Ω). Furthermore, if p = n or s < p∗ , then kuk∞ ≤ M < ∞, with the constant M depending only on b1 , Ω and k(u − b1 )+ k.
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