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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 solutions 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 Hölder 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) satisfying 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 comparison 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. Takáˇc 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: 4 • 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 Poincaré-type inequality, 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} 6 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 Schrödinger operator P we have the following dichotomy: either there exists a strictly positive potential W such that the Schrödinger 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 followingPoincarétype 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) 8 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 infinity 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 minimal 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