On positive solutions of p-Laplacian-type equations
aa r X i v : . [ m a t h . A P ] J a n On positive solutions of p -Laplacian-typeequations Yehuda Pinchover
Department of MathematicsTechnion - Israel Institute of TechnologyHaifa 32000, [email protected]
Kyril Tintarev
Department of MathematicsUppsala UniversitySE-751 06 Uppsala, [email protected]
November 6, 2018
Dedicated to Vladimir Maz’ya on the occasion of his 70th birthday
Abstract
Let Ω be a domain in R d , d ≥
2, and 1 < p < ∞ . Fix V ∈ L ∞ loc (Ω).Consider the functional Q and its Gˆateaux derivative Q ′ given by Q ( u ) :=1 p Z Ω ( |∇ u | p + V | u | p )d x, Q ′ ( u ) := −∇ · ( |∇ u | p − ∇ u ) + V | u | p − u. In this paper we discuss several aspects of relations between functional-analytic properties of the functional Q and properties of positive so-lutions of the equation Q ′ ( u ) = 0.2000 Mathematics Subject Classification.
Primary 35J60; Secondary35J20, 35J70, 49R50.
Keywords. quasilinear elliptic operator, p -Laplacian, ground state,positive solutions, comparison principle, minimal growth. Properties of positive solutions of quasilinear elliptic equations, and in partic-ular of equations with the p -Laplacian term in the principal part, have been1xtensively studied over the recent decades, (see for example [3, 4, 28, 32]and the references therein). Fix p ∈ (1 , ∞ ), a domain Ω ⊆ R d and a realvalued potential V ∈ L ∞ loc (Ω). The p -Laplacian equation in Ω with potential V is the equation of the form − ∆ p ( u ) + V | u | p − u = 0 in Ω , (1.1)where ∆ p ( u ) := ∇· ( |∇ u | p − ∇ u ) is the celebrated p -Laplacian. This equation,in the semistrong sense, is a critical point equation for the functional Q ( u ) = Q V ( u ) := 1 p Z Ω ( |∇ u | p + V | u | p ) d x u ∈ C ∞ (Ω) . (1.2)So, we consider solutions of (1.1) in the following weak sense. Definition 1.1.
A function v ∈ W ,p loc (Ω) is a (weak) solution of the equation Q ′ ( u ) := − ∆ p ( u ) + V | u | p − u = 0 in Ω , (1.3)if for every ϕ ∈ C ∞ (Ω) Z Ω ( |∇ v | p − ∇ v · ∇ ϕ + V | v | p − vϕ ) d x = 0 . (1.4)We say that a real function v ∈ C (Ω) is a supersolution (resp. subsolution )of the equation (1.3) if for every nonnegative ϕ ∈ C ∞ (Ω) Z Ω ( |∇ v | p − ∇ v · ∇ ϕ + V | v | p − vϕ ) d x ≥ ≤ local properties of solutions of (1.3) that hold in anysmooth subdomain Ω ′ ⋐ Ω (i.e., Ω ′ is compact in Ω).
1. Smoothness and Harnack inequality.
Weak solutions of (1.3) admitH¨older continuous first derivatives, and nonnegative solutions of (1.3) satisfythe Harnack inequality (see for example [14, 34, 35, 32, 38]).
2. Principal eigenvalue and eigenfunction.
For any smooth subdomainΩ ′ ⋐ Ω consider the variational problem λ ,p (Ω ′ ) := inf u ∈ W ,p (Ω ′ ) R Ω ′ ( |∇ u | p + V | u | p ) d x R Ω ′ | u | p d x . (1.6)2t is well-known that for such a subdomain, (1.6) admits (up to a multiplica-tive constant) a unique minimizer ϕ [9, 13]. Moreover, ϕ is a positive solutionof the quasilinear eigenvalue problem ( Q ′ ( ϕ ) = λ ,p (Ω ′ ) | ϕ | p − ϕ in Ω ′ ,ϕ = 0 on ∂ Ω ′ . (1.7) λ ,p (Ω ′ ) and ϕ are called, respectively, the principal eigenvalue and eigen-function of the operator Q ′ in Ω ′ .
3. Weak and strong maximum principles.Theorem 1.2 ([13] (see also [3, 4])) . Assume that Ω ⊂ R d is a bounded C α -domain, where < α ≤ . 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 ≥ in Ω with some f ∈ L ∞ (Ω) , and satisfies u ≥ on ∂ Ω ,then u is nonnegative in Ω .(ii) Q ′ satisfies the strong maximum principle: If u is a solution of theequation Q ′ ( u ) = f (cid:9) in Ω with some f ∈ L ∞ (Ω) , and satisfies u ≥ on ∂ Ω , then u > in Ω .(iii) λ ,p (Ω) > .(iv) For some (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 ∈ W ,p (Ω) ∩ L ∞ (Ω) satis-fying Q ′ ( v ) = f (cid:9) in Ω , such that v | ∂ Ω ∈ C α ( ∂ Ω) and f ∈ L ∞ (Ω) .(v) For each nonnegative f ∈ C α (Ω) ∩ L ∞ (Ω) there exists a unique weaknonnegative 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 ⊂ R d be a bounded domain of class C ,α , where < α ≤ , and suppose that V ∈ L ∞ (Ω) . Assume that λ ,p (Ω) > and let i ∈ W ,p (Ω) ∩ L ∞ (Ω) satisfying Q ′ ( u i ) ∈ L ∞ (Ω) , u i | ∂ Ω ∈ C α ( ∂ Ω) , where i = 1 , . Suppose further that the following inequalities are satisfied Q ′ ( u ) ≤ Q ′ ( u ) in Ω ,Q ′ ( u ) ≥ in Ω ,u ≤ u on ∂ Ω ,u ≥ on ∂ Ω . (1.8) Then u ≤ u in Ω .
5. Strong comparison principle.Definition 1.4. we say that the strong comparison principle (or SCP forbrevity) holds true for the functional Q if the conditions of Theorem 1.3implies that u < u in Ω unless u = u in Ω. 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 SCPsee [2, 5, 7, 16, 32, 37] and the references therein. In [5] M. Cuesta andP. Tak´aˇc present a counterexample where the WCP holds true but the SCPdoes not.Throughout this paper we assume that Q ( u ) ≥ ∀ u ∈ C ∞ (Ω) . (1.9)The following Allegretto-Piepenbrink-type theorem, links the existence ofpositive 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) .(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-analyticproperties of the functional Q and properties of its positive solutions. Inparticular, we review the following topics:4 A representation of the nonnegative functional Q as an integral of anonnegative Lagrangian density and the existence of a useful equivalentnonnegative 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 interms of the variational capacity of balls (Section 3). • The identification of ground state as a global minimal solution (Section4). • A theorem of Liouville type connecting the behavior of a ground state ofone functional with the existence of a ground state of another functionalwith a given ‘decaying’ subsolution (Section 5). • A variational principle that characterizes solutions of minimal growthat 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, thedefinition of a natural functional space associated with the functional Q (Section 9). Let v ∈ C (Ω) be a positive solution (resp. subsolution) of (1.3). Usingthe positive Lagrangian representation from [3, 4, 6], we infer that for every u ∈ C ∞ (Ω), u ≥ Q ( u ) = Z Ω L ( u, v ) d x, resp. Q ( u ) ≤ Z Ω L ( u, v ) d x, (2.1)5here L ( u, v ) := 1 p (cid:20) |∇ u | p + ( p − u p v p |∇ v | p − p u p − v p − ∇ u · |∇ v | p − ∇ v (cid:21) . (2.2)It can be easily verified that L ( u, v ) ≥ w := u/v , where v is a positive solution of (1.3) and u ∈ C ∞ (Ω), u ≥
0. Then (2.1) implies that Q ( vw ) = 1 p Z Ω (cid:2) | w ∇ v + v ∇ w | p − w p |∇ v | p − pw p − v |∇ v | p − ∇ v · ∇ w (cid:3) d x. (2.3)Similarly, if v is a nonnegative subsolution of (1.3), then Q ( vw ) ≤ p Z Ω (cid:2) | w ∇ v + v ∇ w | p − w p |∇ v | p − pw p − v |∇ v | p − ∇ v · ∇ w (cid:3) d x. (2.4)Therefore, a nonnegative functional Q can be represented as the integral ofa 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 posesobvious difficulties for extending the domain of the functional to more generalweakly differentiable functions. The next proposition shows that Q admitsa two-sided estimate by a simplified Lagrangian containing only nonnegativeterms. 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 thereexists a positive constant C such that C − g ≤ f ≤ Cg . Proposition 2.1 ([26, Lemma 2.2]) . Let v ∈ C (Ω) be a positive solutionof (1.3). Then Q ( vw ) ≍ Z Ω v |∇ w | ( w |∇ v | + v |∇ w | ) p − d x ∀ w ∈ C (Ω) , w ≥ . (2.5) In particular, for p ≥ , we have Q ( vw ) ≍ Z Ω (cid:0) v p |∇ w | p + v |∇ v | p − w p − |∇ w | (cid:1) d x ∀ w ∈ C (Ω) , w ≥ . (2.6) If v is only a nonnegative subsolution of (1.3), then for < p < ∞ wehave Q ( vw ) ≤ C Z Ω ∩{ v> } v |∇ w | ( w |∇ v | + v |∇ w | ) p − d x ∀ w ∈ C (Ω) , w ≥ . n particular, for p ≥ we have Q ( vw ) ≤ C Z Ω (cid:0) v p |∇ w | p + v |∇ v | p − w p − |∇ w | (cid:1) d x ∀ w ∈ C (Ω) , w ≥ . Remark 2.2.
It is shown in [26] that for p >
It is well known (see [21]) that for a nonnegative Schr¨odinger operator P wehave the following dichotomy: either there exists a strictly positive potential W such that the Schr¨odinger operator P − W is nonnegative, or P admitsa unique (generalized) ground state. It turns out that this statement is alsotrue for nonnegative functionals of the form (1.2). Definition 3.1.
Let Q be a nonnegative functional on C ∞ (Ω) of the form(1.2). We say that a sequence { u k } ⊂ C ∞ (Ω) of nonnegative functions is a null sequence of the functional Q in Ω, if there exists an open set B ⋐ Ω suchthat R B | u k | p d x = 1, andlim k →∞ Q ( u k ) = lim k →∞ Z Ω ( |∇ u k | p + V | u k | p ) d x = 0 . (3.1)We say that a positive function v ∈ C (Ω) is a ground state of the functional Q in Ω if v is an L p loc (Ω) limit of a null sequence of Q . If Q ≥
0, and Q admitsa ground state in Ω, we say that Q is critical in Ω. Remark 3.2.
The requirement that { u k } ⊂ C ∞ (Ω), can clearly be weakenedby assuming only that { u k } ⊂ W ,p (Ω). Also, as it follows from Theorem 3.4,the requirement R B | u k | p d x = 1 can be replaced by R B | u k | p d x ≍ R B u k d x ≍ p = 2). Theorem 3.3.
Suppose that the functional Q is nonnegative on C ∞ (Ω) . . 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 positivesupersolution.3. Q is critical in Ω if and only if Q admits a null sequence that convergeslocally uniformly in Ω .4. If Q admits a ground state v , then the following Poincar´e type inequalityholds: There exists a positive continuous function W in Ω , such thatfor every ψ ∈ C ∞ (Ω) satisfying R ψv d x = 0 there exists a constant C > such that the following inequality holds: Q ( u ) + C (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψu d x (cid:12)(cid:12)(cid:12)(cid:12) p ≥ C − Z Ω W ( |∇ u | p + | u | p ) d x ∀ u ∈ C ∞ (Ω) . (3.2)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) . Thefollowing statements are equivalent.(a) Q does not admit a ground state in Ω .(b) There exists a continuous function W > in Ω such that Q ( u ) ≥ Z Ω W ( x ) | u ( x ) | p d x ∀ u ∈ C ∞ (Ω) . (3.3) (c) There exists a continuous function W > in Ω such that Q ( u ) ≥ Z Ω W ( x ) ( |∇ u ( x ) | p + | u ( x ) | p ) d x ∀ u ∈ C ∞ (Ω) . (3.4) (d) There exists an open set B ⋐ Ω and C B > such that Q ( u ) ≥ C B (cid:12)(cid:12)(cid:12)(cid:12)Z B u ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) p ∀ u ∈ C ∞ (Ω) . (3.5)8 uppose further that d > p . Then Q does not admit a ground state in Ω ifand only if there exists a continuous function W > in Ω such that Q ( u ) ≥ (cid:18)Z Ω W ( x ) | u ( x ) | p ∗ d x (cid:19) p/p ∗ ∀ u ∈ C ∞ (Ω) , (3.6) 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)which is not critical is said to be subcritical (or weakly coercive ) in Ω. Example 3.6.
Consider the functional Q ( u ) := R R d |∇ u | p d x . It follows from[20, Theorem 2] that if d ≤ p , then Q admits a ground state ϕ = constant in R d . On the other hand, if d > p , then u ( x ) := (cid:2) | x | p/ ( p − (cid:3) ( p − d ) /p , v ( x ) := constantare two positive supersolutions of the equation − ∆ p u = 0 in R d . Therefore, Q is weakly coercive in R d . Example 3.7.
Let d > d = p , and Ω := R d \ { } be the punctured space.The following celebrated Hardy’s inequality holds true: Q λ ( u ) := Z Ω (cid:18) |∇ u | p − λ | u | p | x | p (cid:19) d x ≥ u ∈ C ∞ (Ω) , (3.7)whenever λ ≤ c ∗ p,d := | ( p − d ) /p | p . Clearly, if λ < c ∗ p,d , then Q λ ( u ) is weaklycoercive. On the other hand, the proof of Theorem 1.3 in [31] shows that Q λ with λ = c ∗ p,d admits a null sequence. It can be easily checked that thefunction v ( r ) := | r | ( p − d ) /p is a positive solution of the corresponding radialequation: −| v ′ | p − (cid:20) ( p − v ′′ + d − r v ′ (cid:21) − c ∗ p,d | v | p − vr p = 0 r ∈ (0 , ∞ ) . Therefore, ϕ ( x ) := | x | ( p − d ) /p is the ground state of the equation − ∆ p u − c ∗ p,d | u | p − u | x | p = 0 in Ω . (3.8)Note that ϕ W ,p loc ( R d ) for p = d . In particular, ϕ is not a positive superso-lution of the equation ∆ p u = 0 in R d .9n [39, 40] Troyanov has established a relationship between the p -capacityof closed balls in a Riemannian manifold M and the p -parabolicity of M withrespect 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).Let K ⋐ Ω be a compact set. The Q -capacity of K in Ω is defined byCap Q ( K, Ω) := inf { Q ( u ) | u ∈ C ∞ (Ω) , u ≥ K } . Corollary 3.9.
Suppose that the functional Q is nonnegative on C ∞ (Ω) .Then Q is critical in Ω if and only if the Q -capacity of each closed ball in Ω is zero. Definition 4.1.
Let K be a compact set in Ω. A positive solution u ofthe equation Q ′ ( u ) = 0 in Ω \ K is said to be a positive solution of minimalgrowth in a neighborhood of infinity in Ω (or u ∈ M Ω ,K for brevity) if for anycompact set K in Ω, with a smooth boundary, such that K ⋐ int( K ), andany positive supersolution v ∈ C ((Ω \ K ) ∪ ∂K ) of the equation Q ′ ( u ) = 0in 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 hasminimal growth in a neighborhood of infinity in Ω (i.e. u ∈ M Ω , ∅ ) is calleda global minimal solution of the equation Q ′ ( u ) = 0 in Ω. Theorem 4.2 ([29, Theorem 5.1], cf. [28]) . Suppose that < p < ∞ , and Q is nonnegative on C ∞ (Ω) . Then for any x ∈ Ω the equation Q ′ ( u ) = 0 hasa positive solution u ∈ M Ω , { x } . 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 < p < ∞ , and assumethat Q is nonnegative on C ∞ (Ω) . Then Q is subcritical in Ω if and only ifthe equation Q ′ ( u ) = 0 in Ω does not admit a global minimal solution of theequation 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. u of the equation Q ′ ( u ) = 0 in a puncturedneighborhood of x which has a nonremovable singularity at x ∈ R d . With-out loss of generality we may assume that x = 0. If 1 < p ≤ d , then thebehavior of u near an isolated singularity is well understood. Indeed, due toa result of L. V´eron (see [28, Lemma 5.1]), we have that u ( x ) ∼ ( | x | α ( d,p ) p < d, − log | x | p = d, as x → , (4.1)where α ( d, p ) := ( p − d ) / ( p − f ∼ g means thatlim x → f ( x ) g ( x ) = C for some positive constant C . In particular, lim x → u ( x ) = ∞ .Assume now that p > d . A general question is whether in this case, anypositive solution of the equation Q ′ ( u ) = 0 in a punctured ball centered at x can be continuously extended at x (see [17] for partial results).Under the assumption that u ≍ Lemma 4.4.
Assume that p > d , and let v be a positive solution of theequation Q ′ ( u ) = 0 in a punctured neighborhood of x satisfying u ≍ near x . Then u can be continuously extended at x . The following statement combines the second part of [28, Theorem 5.4],where the case 1 < p ≤ d is considered with [29, Theorem 5.3] which dealswith the case p > d . Theorem 4.5.
Let x ∈ Ω , and let u ∈ M Ω , { x } . Then Q is subcritical in Ω if and only if u has a nonremovable singularity at x . In [26] we use some of the positivity properties of the nonnegative functional Q discussed in the previous sections to prove a Liouville comparison principlefor equations Q ′ ( u ) = 0 in Ω. (see Theorem 9.1 for the case p = 2).11 heorem 5.1. Let Ω be a domain in R d , d ≥ , and let p ∈ (1 , ∞ ) . For j = 0 , , let V j ∈ L ∞ loc (Ω) , and let Q j ( u ) := Z Ω ( |∇ u ( x ) | p + V j ( x ) | u ( x ) | p ) d x u ∈ C ∞ (Ω) . Assume that the following assumptions hold true.(i) The functional Q admits a ground state ϕ in Ω .(ii) Q ≥ on C ∞ (Ω) , and the equation Q ′ ( u ) = 0 in Ω admits a subsolu-tion ψ ∈ W ,p loc (Ω) satisfying ψ + = 0 , where ψ + ( x ) := max { , ψ ( x ) } .(iii) The following inequality holds in Ω ψ + ≤ Cϕ, (5.1) where
C > is a positive constant.(iv) The following inequality holds in Ω |∇ ψ + | p − ≤ C |∇ ϕ | p − , (5.2) where C > is a positive constant.Then the functional Q admits a ground state in Ω , and ψ is the groundstate. In particular, ψ is (up to a multiplicative constant) the unique positivesupersolution of the equation Q ′ ( u ) = 0 in Ω . Remark 5.2.
Condition (5.2) is redundant for p = 2. For p = 2 it isequivalent to the assumption that the following inequality holds in Ω: ( |∇ ψ + | ≤ C |∇ ϕ | if p > , |∇ ψ + | ≥ C |∇ ϕ | if p < , (5.3)where C >
Remark 5.3.
This theorem holds if, in addition to (5.1), one assumes insteadof |∇ ψ + | p − ≤ C |∇ ϕ | p − in Ω (see (5.2)), that the following inequality holdstrue in Ω ψ |∇ ψ + | p − ≤ Cϕ |∇ ϕ | p − , (5.4)where C > emark 5.4.
Suppose that 1 < p <
2, and assume that the ground state ϕ > Q is such that w = is a ground state of thefunctional E ϕ ( w ) = Z Ω ϕ p |∇ w | p d x, (5.5)that is, there is a sequence { w k } ⊂ C ∞ (Ω) of nonnegative functions satisfying E ϕ ( w k ) →
0, and R B | w k | p = 1 for a fixed B ⋐ Ω (this implies that w k → in L p loc (Ω)). In this case, the conclusion of Theorem 5.1 holds if there is anonnegative subsolution ψ + of Q ′ ( u ) = 0 satisfying (5.1) alone, without anassumption on the gradients (like (5.2) or (5.4)). Remark 5.5.
Condition (5.2) is essential when p >
2, and presumablyalso when p <
2. When p >
2, Ω = R d and V is radially symmetric,Proposition 4.2 in [26] shows that the simplified energy functional is notequivalent to either of its two terms that lead to conditions (5.1) and (5.2),respectively. Example 5.6.
Assume that 1 ≤ d ≤ p ≤ p >
1, Ω = R d , and consider thefunctional Q ( u ) := R R d |∇ u | p d x . By Example 3.6, the functional Q admitsa ground state ϕ = constant in R d .Let Q be a functional of the form (1.2) satisfying Q ≥ C ∞ ( R d ).Let ψ ∈ W ,p loc ( R d ), ψ + = 0 be a subsolution of the equation Q ′ ( u ) = 0 in R d ,such that ψ + ∈ L ∞ ( R d ). It follows from Theorem 5.1 that ψ is the groundstate of Q in R d . In particular, ψ is (up to a multiplicative constant) theunique positive supersolution and unique bounded solution of the equation Q ′ ( u ) = 0 in R d . Note that there is no assumption on the behavior ofthe potential V at infinity. This result generalizes some striking Liouvilletheorems for Schr¨odinger operators on R d that hold for d = 1 , p = 2(see [24, theorems 1.4–1.6]). Example 5.7.
Let 1 < p < ∞ , d > d = p , and Ω := R d \ { } be thepunctured space. Let Q be a functional of the form (1.2) satisfying Q ≥ C ∞ (Ω). Let ψ ∈ W ,p loc (Ω), ψ + = 0 be a subsolution of the equation Q ′ ( u ) = 0 in Ω, satisfying ψ + ( x ) ≤ C | x | ( p − d ) /p x ∈ Ω . (5.6)When p >
2, we require in addition that the following inequality is satisfied ψ + ( x ) |∇ ψ + ( x ) | p − ≤ C | x | − d x ∈ Ω . (5.7)13t follows from Theorem 5.1, Remark 5.3, Remark 5.4 and Example 3.7 that ψ is the ground state of Q in Ω. Let ϕ be the ground state of the Hardyfunctional. The reason that (5.7) is stated only for p > < p < E ϕ ( w ) = Z Ω | x | p − d |∇ w | p d x (5.8)admits a ground state , so Remark 5.4 applies.Next, we present a family of functionals Q for which the conditions ofExample 5.7 are satisfied. Example 5.8.
Let d ≥
2, 1 < p < d , α ≥
0, and Ω := R d \ { } . Let W α ( x ) := − (cid:18) d − pp (cid:19) p αdp/ ( d − p ) + | x | pp − (cid:16) α + | x | pp − (cid:17) p . Note that if α = 0 this is the Hardy potential as in Example 3.7. If Q is thefunctional (1.2) with the potential V := W α , then ψ α ( x ) := (cid:16) α + | x | pp − (cid:17) − ( d − p )( p − p is a solution of Q ′ ( u ) = 0 in Ω, and therefore Q ≥ C ∞ (Ω). Moreover,one can use Example 5.7 to show that ψ α is a ground state of Q . Note firstthat ψ = ψ α satisfies (5.6). If dd − < p < d , then ψ α satisfies also (5.7) andtherefore, it is a ground state in this case. In the remaining case p ≤ dd − ≤ ψ α is a ground state from the property of thefunctional (5.8). The aim of this section is to represent positive solutions of minimal growthin a neighborhood of infinity in Ω as a limit of a modified null sequence.
Theorem 6.1 ([29, Theorem 7.1]) . Suppose that < p < ∞ , and let Q V benonnegative on C ∞ (Ω) . Let Ω ⋐ Ω be an open set, and let u ∈ C (Ω \ Ω ) be positive solution of the equation Q ′ V ( u ) = 0 in Ω \ Ω satisfying |∇ u | 6 = 0 in Ω \ Ω .Then u ∈ M Ω , Ω if for every smooth open set Ω satisfying Ω ⋐ Ω ⋐ Ω ,and an open set B ⋐ (Ω \ Ω ) there exists a sequence { u k } ⊂ C ∞ (Ω) , u k ≥ ,such that for all k ∈ N , R B | u k | p d x = 1 , and lim k →∞ Z Ω \ Ω L ( u k , u ) d x = 0 , (6.1) where L is the Lagrangian given by (2.2) . Conjecture 6.2.
We conjecture that for p = 2 a positive global solution ofthe equation Q ′ V ( u ) = 0 in Ω satisfying u ∈ M Ω , Ω for some smooth open setΩ ⋐ Ω is a global minimal solution.
Remark 6.3.
The validity of Conjecture 6.2 seems to be related to the SCP.We note that if Conjecture 6.2 holds true, then the condition of Theorem 6.1is also necessary (cf. Section 9.3).Finally, we formulate a sub-supersolution comparison principle for oursingular elliptic equation.
Theorem 6.4 (Comparison Principle [29]) . Assume that the functional Q V is nonnegative on C ∞ (Ω) . Fix smooth open sets Ω ⋐ Ω ⋐ Ω . Let u, v ∈ W ,p loc (Ω \ Ω ) ∩ C (Ω \ Ω ) be, respectively, a positive subsolution and a super-solution of the equation Q ′ V ( w ) = 0 in Ω \ Ω such that u ≤ v on ∂ Ω .Assume further that Q ′ V ( u ) ∈ L ∞ loc (Ω \ Ω ) , |∇ u | 6 = 0 in Ω \ Ω , and thatthere exist an open set B ⋐ (Ω \ Ω ) and a sequence { u k } ⊂ C ∞ (Ω) , u k ≥ ,such that Z B | u k | p d x = 1 ∀ k ≥ , and lim k →∞ Z Ω \ Ω L ( u k , u ) d x = 0 . (6.2) Then u ≤ v on Ω \ Ω . In this section we discuss some results of [36] concerning the solvability ofthe nonhomogeneous equation Q ′ V ( u ) = f in Ω , (7.1)15here Q V is the nonnegative functional (1.2). In some cases, e.g. V ≥ p = 2, the nonnegativity of Q V implies that Q V is convex. In general, how-ever, Q V might be nonconvex. For p >
2, see the elementary one-dimensionalexample at the end of [8], and also the proof of [13, Theorem 7]. For p < Q V is convex and weakly coercive, then (7.1) can be easily solved bydefining a Banach space X as a completion of C ∞ (Ω) with respect to thenorm Q V ( · ) /p (see the discussion of the analogous space for p = 2 in Section 9below). Such space is continuously imbedded into W ,p loc (Ω) by (3.4). It followsthat for every f ∈ X ∗ the functional u Q V ( u ) − h f, u i u ∈ X, has a minimum that solves (7.1).Note that the requirement of weak coercivity cannot be removed. Indeed,if p = 2, Ω is smooth and bounded, and V = 0, then the correspondingground state ϕ is the first eigenfunction of the Dirichlet Laplacian with aneigenvalue λ , and there is no solution u ∈ W , (Ω) to the equation( − ∆ − λ ) u = f in Ω (7.2)unless R Ω ϕ ( x ) f ( x ) d x = 0.In order to address the nonconvex case, we use the following setup fromconvex analysis (see [10, Chapt. I] for details.) The polar (or conjugate )functional to Q V is defined by Q ∗ V ( f ) := sup u ∈ C ∞ (Ω) [ h u, f i − Q V ( u )] f ∈ D ′ (Ω) . (7.3)Notice that Q V is positively homogeneous of degree p , and consequently, Q ∗ V is positively homogeneous functional of degree p ′ , where p ′ = p/ ( p − effective or natural ) domain X ∗ of Q ∗ V is defined naturally by X ∗ = { f ∈ D ′ (Ω) : Q ∗ V ( f ) < ∞} . (7.4)The definition of Q ∗ V ( f ) in (7.3) yields immediately the well known Fenchel-Young inequality |h u, f i| ≤ Q V ( u ) + Q ∗ V ( f ) , (7.5)and equivalently, the H¨older inequality |h u, f i| ≤ ( p Q V ( u )) /p ( p ′ Q ∗ V ( f )) /p ′ . (7.6)16ne can easily verify that X ∗ is a linear subspace of D ′ (Ω) and k f k ∗ := ( p ′ Q ∗ V ( f )) /p ′ (7.7)defines a norm on X ∗ . In particular, k f k ∗ = 0 implies f = 0, as a consequenceof (7.6) combined with the separation property of the duality between C ∞ (Ω)and D ′ (Ω).From (3.4) one immediately deduces that X ∗ contains (cid:0) W ,p (Ω; W ) (cid:1) ′ andthat the corresponding embedding (cid:0) W ,p (Ω; W ) (cid:1) ′ ֒ → X ∗ is continuous and dense. The density follows from C ∞ (Ω) ⊂ (cid:0) W ,p (Ω; W ) (cid:1) ′ ⊂ X ∗ ⊂ D ′ (Ω) . (7.8)Therefore, denoting by X ∗∗ the (strong) dual space of X ∗ with respectto the duality h · , · i , we observe that X ∗∗ is continuously embedded into W ,p (Ω; W ) and that C ∞ (Ω) is weak-star dense in X ∗∗ . It is noteworthythat the separability of X ∗ in the norm topology implies that the weak-startopology on any bounded subset of X ∗∗ is metrizable (Rudin [33, Theorem3.16, p. 70]). Now consider the bipolar (or second conjugate ) functional to Q V defined by Q ∗∗ V ( u ) := sup f ∈ X ∗ [ h u, f i − Q ∗ V ( f )] u ∈ X ∗∗ . (7.9)From (7.5) it is evident that0 ≤ Q ∗∗ V ( u ) ≤ Q V ( u ) for every u ∈ C ∞ (Ω) . (7.10)Moreover, in analogy with the norm k · k ∗ on X ∗ (see (7.7)), the dual norm k · k ∗∗ on X ∗∗ is given by k u k ∗∗ = ( p Q ∗∗ V ( u )) /p . (7.11)The Fenchel-Young and H¨older inequalities, (7.5) and (7.6), respectively,remain valid with Q ∗∗ V ( u ) in place of Q V . In particular, we have |h u, f i| ≤ k u k ∗∗ k f k ∗ ≤ ( p Q V ( u )) /p k f k ∗ for f ∈ X ∗ , u ∈ C ∞ (Ω) . (7.12)It follows [10, Ch. I, § f ∈ X ∗ ,inf u ∈ C ∞ (Ω) [ Q V ( u ) − h u, f i ] = inf u ∈ C ∞ (Ω) [ Q ∗∗ V ( u ) − h u, f i ] = − Q ∗ V ( f ) . (7.13)17 efinition 7.1. Given a distribution f ∈ X ∗ , we say that a function u ∈ W ,p loc (Ω) is a generalized (or relaxed ) minimizer for the functional u Q V ( u ) − h u, f i if it has the following three properties:(i) u is a (true) minimizer for the functional u Q ∗∗ V ( u ) − h u, f i : X ∗∗ → R , hence, u ∈ X ∗∗ .(ii) u satisfies equation Q ′ V ( u ) = f in the sense of distributions on Ω.(iii) There exists a (minimizing) sequence { u k } ∞ k =1 ⊂ C ∞ (Ω) such that Q V ( u k ) − h u k , f i → − Q ∗ V ( f ) and Q ′ V ( u k ) − f ≡ Q ′ V ( u k ) − h · , f i → W − ,p ′ loc (Ω) (7.14)as k → ∞ , together with u k → u strongly in W ,p loc (Ω), and u k ∗ ⇀ u weakly-star in X ∗∗ as k → ∞ .We can now formulate the existence result. Theorem 7.2 ([36, Theorem 4.3]) . Let Ω ⊂ R d be a domain, < p < ∞ , and V ∈ L ∞ loc (Ω) . Assume that the nonnegative functional Q V is weakly coercive.Then, for every f ∈ X ∗ , the functional u Q V ( u ) − h u, f i is bounded frombelow on C ∞ (Ω) and has a generalized minimizer u in X ∗∗ ( ⊂ W ,p loc (Ω) ).In particular, this minimizer verifies the equation Q ′ V ( u ) = f . In this section we discuss positivity properties of the functional Q from[28, 30] along the lines of criticality theory for second-order linear ellipticoperators [22, 23]. We note that Theorem 8.6 and Example 8.7 are newresults. Proposition 8.1.
Let V i ∈ L ∞ loc (Ω) . If V (cid:9) V and Q V ≥ , then Q V issubcritical (weakly coercive). Proposition 8.2.
Let Ω ⊂ Ω be domains in R d such that Ω \ Ω = ∅ . Let Q V be defined on C ∞ (Ω ) .1. If Q V ≥ on C ∞ (Ω ) , then Q V is subcritical in Ω .2. If Q V is critical in Ω , then Q V is nonpositive in Ω . roposition 8.3. Let V , V ∈ L ∞ loc (Ω) , V = V . For t ∈ R we denote Q t ( u ) := tQ V ( u ) + (1 − t ) Q V ( u ) , (8.1) and suppose that Q V i ≥ on C ∞ (Ω) for i = 0 , .Then Q t ≥ on C ∞ (Ω) for all t ∈ [0 , . Moreover, Q t is subcritical in Ω for all t ∈ (0 , . Proposition 8.4.
Let Q V be a subcritical functional in Ω . Consider V ∈ L ∞ (Ω) such that V (cid:3) and supp V ⋐ Ω . Then there exist τ + > and −∞ ≤ τ − < such that Q V + tV is subcritical in Ω for t ∈ ( τ − , τ + ) , and Q V + τ + V is critical in Ω . Proposition 8.5.
Assume that Q V admits a ground state v in Ω . Consider V ∈ L ∞ (Ω) such that supp V ⋐ Ω . Then there exists < τ + ≤ ∞ such that Q V + tV is subcritical in Ω for t ∈ (0 , τ + ) if and only if Z Ω V | v | p d x > . (8.2)In propositions 8.4 and 8.5 we assumed that the perturbation V has acompact support. In the following we consider a wider class of perturbations.Assume that Q is subcritical in Ω and d > p . It follows from Theorem 3.4that there exists a continuous weight function W such that the followingHardy-Sobolev-Maz’ya inequality is satisfied Q ( u ) ≥ (cid:18)Z Ω W ( x ) | u ( x ) | p ∗ d x (cid:19) p/p ∗ ∀ u ∈ C ∞ (Ω) , (8.3)where p ∗ = pd/ ( d − p ) is the critical Sobolev exponent.The following theorem shows that for a certain class of potentials ˜ V theabove Hardy-Sobolev-Maz’ya inequality is preserved with the same weightfunction W . This theorem extends the analogous result for p = 2 proved in[30, Theorem 2.9]. We may say that such ˜ V are small perturbations of thefunctional Q in Ω. Theorem 8.6.
Let Q be the functional (1.2) with d > p and suppose that Q ( u ) ≥ (cid:18)Z Ω W | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω) , (8.4)19 here W is some positive continuous function (see Theorem 3.4, and in par-ticular (3.6) ). Let ˜ V ∈ L ∞ loc (Ω) ∩ L d/p (Ω; W − d/p ∗ ) . (8.5) Consider the one-parameter family of functionals ˜ Q λ defined by ˜ Q λ ( u ) := Q ( u ) + λ Z Ω ˜ V | u | p d x u ∈ C ∞ (Ω) , where λ ∈ R , and let S := { λ ∈ R | ˜ Q λ ≥ on C ∞ (Ω) } . (i) If λ ∈ S and ˜ Q λ is subcritical in Ω , then there exists C > such that ˜ Q λ ( u ) ≥ C (cid:18)Z Ω W | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω) . (8.6) (ii) If λ ∈ S and ˜ Q λ admits a ground state v , then for every ψ ∈ C ∞ (Ω) such that R Ω ψv d x = 0 there exist C, C > such that the following Hardy-Sobolev-Maz’ya-Poincar´e type inequality holds ˜ Q λ ( u ) + C (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψu d x (cid:12)(cid:12)(cid:12)(cid:12) p ≥ C (cid:18)Z Ω W | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω) . (8.7) (iii) The set S is a closed interval with a nonempty interior which isbounded if and only if ˜ V changes its sign on a set of a positive measure in Ω . Moreover, λ ∈ ∂S if and only if ˜ Q λ is critical in Ω .Proof. (i)–(ii) Assume first that ˜ Q λ is subcritical in Ω, and (8.6) does nothold, then there exists a sequence { u k } ⊂ C ∞ (Ω) of nonnegative functionssuch that ˜ Q λ ( u k ) →
0, and R Ω W | u k | p ∗ d x = 1. In light of (3.4) (with anotherweight function ˜ W ) it follows that that u k → W ,p loc (Ω).If ˜ Q λ has a ground state v , and (8.7) does not hold. Then there existsa sequence { u k } ⊂ C ∞ (Ω) of nonnegative functions such that ˜ Q λ ( u k ) → R Ω ψu k d x →
0, but R Ω W | u k | p ∗ d x = 1. It follows from (3.2) (with anotherweight function ˜ W ) that u k → W ,p loc (Ω).Consequently, for any K ⋐ Ω we havelim k →∞ Z K | ˜ V || u k | p d x = 0 . (8.8)20n the other hand, (8.5) and H¨older inequality imply that for any ε > K ε ⋐ Ω such that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω \ K ε | ˜ V || u k | p d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z Ω \ K ε | ˜ V | d/p W − d/p ∗ d x (cid:19) p/d (cid:18)Z Ω W | u k | p ∗ d x (cid:19) p/p ∗ < ε. (8.9)Therefore, R Ω | ˜ V || u k | p d x →
0. Since Q ( u k ) ≤ ˜ Q λ ( u k ) + | λ | Z Ω | ˜ V || u k | p d x, (8.10)it follows that Q ( u k ) →
0. Hence, (8.4) implies that R Ω W | u k | p ∗ d x → R Ω W | u k | p ∗ d x = 1. Consequently, (8.6)(resp. (8.7)) holds true.(iii) It follows from Proposition 8.3 that S is an interval, and that λ ∈ int S implies that ˜ Q λ is subcritical in Ω. The claim on the boundedness of S istrivial and left to the reader.On the other hand, suppose that for some λ ∈ R the functional ˜ Q λ issubcritical. By part (i), ˜ Q λ satisfies (8.6) with weight W . Therefore, (8.9)(with K ε = ∅ ) implies that˜ Q λ ( u ) ≥ C (cid:18)Z Ω W | u | p ∗ d x (cid:19) p/p ∗ ≥ C (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ˜ V | u | p d x (cid:12)(cid:12)(cid:12)(cid:12) u ∈ C ∞ (Ω) . (8.11)Therefore, λ ∈ int S . Consequently, λ ∈ ∂S implies that ˜ Q λ is critical in Ω.In particular, 0 ∈ int S . Example 8.7.
Let 2 ≤ p < d , and let Ω ⊂ R d be a bounded convex domainwith a smooth boundary. Consider the Hardy functional Q ( u ) := Z Ω |∇ u | p d x − (cid:12)(cid:12)(cid:12)(cid:12) p − p (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω | u | p d ( x, ∂ Ω) p d x. By [11], the functional Q satisfies the following Hardy-Sobolev-Maz’yatype inequality Q ( u ) ≥ C (cid:18)Z Ω | u | p ∗ d x (cid:19) p/p ∗ . (8.12)21et V ∈ L ∞ loc (Ω) ∩ L d/p (Ω) be a positive function. By [19], there exists aconstant λ ∗ > Q λ ( u ) := Q ( u ) − λ Z Ω V | u | p d x is nonnegative for all λ ≤ λ ∗ . Now, Theorem 8.6 implies that the functional Q λ ∗ ( u ) is critical in Ω. Moreover, for λ < λ ∗ the following inequality holds Q λ ( u ) ≥ C λ (cid:18)Z Ω | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω)for some C λ > ψ ∈ C ∞ (Ω) the followinginequality holds Q λ ∗ ( u ) + C (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uψ d x (cid:12)(cid:12)(cid:12)(cid:12) p ≥ C (cid:18)Z Ω | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω)for some C, C > p = 2 ) Some of the results in the preceding sections have stronger counterparts inthe linear case (for a recent review on the theory of positive solutions ofsecond-order linear elliptic PDEs, see [25] and the references therein). Inparticular, there are several properties that are true for the linear case butare generally false or unknown in the general case. This refers in particularto SCP whose scope of validity when p = 2 is not completely understood,and to the convexity of the functional Q , which is known to be generallyfalse for p > Q V of the form (1.2).Let A : Ω → R d be a measurable symmetric matrix valued function suchthat for every compact set K ⋐ Ω there exists µ K > µ − K I d ≤ A ( x ) ≤ µ K I d ∀ x ∈ K, (9.1)where I d is the d -dimensional identity matrix, and the matrix inequality A ≤ B means that B − A is a nonnegative matrix on R d . Let V ∈ L q loc (Ω)22e a real potential, where q > d/
2. We consider the quadratic form a A,V [ u ] := 12 Z Ω (cid:0) A ∇ u · ∇ u + V | u | (cid:1) d x (9.2)on C ∞ (Ω) associated with the Schr¨odinger equation P u := ( −∇ · ( A ∇ ) + V ) u = 0 in Ω . (9.3)We say that a A,V is nonnegative on C ∞ (Ω), if a A,V [ u ] ≥ u ∈ C ∞ (Ω).Let v be a positive solution of the equation P u = 0 in Ω. Then by[27, Lemma 2.4] we have the following analog of (2.3). For any nonnegative w ∈ C ∞ (Ω) we have a A,V [ vw ] = 12 Z Ω v A ∇ w · ∇ w d x. (9.4)Moreover, it follows from [27, 28] that all the results mentioned in this paperconcerning the functional Q are also valid for the form a A,V . In the linear case we have the following stronger Liouville-type statement (cf.Theorem 5.1 for p = 2). Theorem 9.1 ([24]) . Let Ω be a domain in R d , d ≥ . Consider two strictlyelliptic Schr¨odinger operators with real coefficients defined on Ω of the form P j := −∇ · ( A j ∇ ) + V j j = 0 , , (9.5) where V j ∈ L p loc (Ω) for some p > d/ , and A j : Ω → R d are measurablesymmetric matrices satisfying (9.1) .Assume that the following assumptions hold true.(i) The operator P admits a ground state ϕ in Ω .(ii) P ≥ on C ∞ (Ω) , and there exists a real function ψ ∈ H (Ω) suchthat ψ + = 0 , and P ψ ≤ in Ω .(iii) The following matrix inequality holds ( ψ + ) ( x ) A ( x ) ≤ Cϕ ( x ) A ( x ) a. e. in Ω , (9.6) where C > is a positive constant. hen the operator P is critical in Ω , and ψ is the corresponding groundstate. In particular, ψ is (up to a multiplicative constant) the unique positivesupersolution of the equation P u = 0 in Ω . D , A,V
When p = 2 and the quadratic form a A,V defined by (9.2) is nonnegative, a A,V induces a scalar product on C ∞ (Ω). One can regard as the naturaldomain of the functional a fA,V ( u ) := a A,V [ u ] − Z Ω uf d x a linear space in which the functional a fA,V has a minimizer for all f suchthat a fA,V is bounded from below. The functional a fA,V is bounded from belowif and only if R Ω uf d x is a continuous functional with respect to the norm( a A,V [ u ]) / . The minimum for such f is not attained on C ∞ (Ω) due to thestrong maximum principle, but any minimizing sequence for a fA,V is a Cauchysequence. Thus, the natural domain of a A,V is the completion of C ∞ (Ω) inthe norm ( a A,V [ u ]) / . In the subcritical case, due to (3.4) (which is valid alsofor subcritical operators of the form (9.3)), this completion is continuouslyimbedded into W , (Ω; W ) for some positive continuous function W . Byanalogy with the classical space D , , we denote the completion space withrespect to the above norm in the subcritical case by D , A,V (Ω) (see [27]).If, however, a A,V has a ground state v , the span of v becomes obviously thezero element of the completion space with respect to the norm ( a A,V [ u ]) / .Recalling the definition of D , ( R d ) for d = 1 ,
2, where the ground state of a I, [ u ] = R R d |∇ u | d x is , and in light of (3.2), we define in the critical casethe norm k u k := a A,V [ u ] + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψu d x (cid:12)(cid:12)(cid:12)(cid:12) ! / , (9.7)where ψ is any C ∞ (Ω)-function satisfying R Ω ψv d x = 0. Hence, also in thecritical case the completion of C ∞ (Ω) with respect to the norm defined by(9.7) (which we also denote by D , A,V (Ω)) is continuously imbedded into thefunction space W , (Ω; W ), with an appropriate weight function W .24 xample 9.2. Let Ω = R d = R n × ( R m \ { } ), 1 ≤ m ≤ d , and denotepoints in Ω by ( x, y ) ∈ R n × ( R m \ { } ). Let a [ u ] = 12 Z Ω |∇ u | d x d y − (cid:18) m − (cid:19) Z Ω u | y | d x d y u ∈ C ∞ (Ω) . (9.8)The functional (9.8) is nonnegative due to the Hardy inequality and theconstant (cid:0) m − (cid:1) is the maximal constant for which this is true. Furthermore,if m < d , the functional (9.8) is weakly coercive, while for m = d it has ageneralized ground state v ( y ) = | y | (2 − m ) / .It follows that completion of C ∞ (Ω) in the norm induced by (9.8) for m < d defines a natural domain for the functional (9.8). It should be noted,however, that on the complete space the integrals in the expression (9.8)might be infinite. A more explicit characterization of the natural domainin this case can be obtained by noting that v ( x, y ) = | y | (2 − m ) / is a positivesolution of the corresponding equation. Thus, for the functional (9.8) thepositive Lagrangian identity (2.1) gives a [ u ] = 12 Z Ω | y | − m |∇ ( | y | ( m − / u ) | d x d y. (9.9)It follows that the completion space can be characterized as a space of mea-surable functions with measurable weak derivatives for which the integral in(9.9) is finite.An analogous definition of the natural domain for the functional Q V couldbe given for general p whenever the functional Q V is convex. We should pointout that while this is in general false, one may require the convexity of anotherfunctional ˆ Q , bounded by Q V from above and from below. In particular, onecan look at the functionals (2.5) or (2.6). The following statement from [30])characterizes a convexity property of ˆ Q given by the right hand side of (2.6). Proposition 9.3.
Let p > , and let v ∈ C (Ω) be a fixed positive function.Then the functional Q ( u ) := Z Ω (cid:2) v p |∇ ( u /p ) | p + v |∇ v | p − u p − /p |∇ ( u /p ) | (cid:3) d x is convex on { u ∈ C ∞ (Ω) , u ≥ } . .3 Positive solutions of minimal growth We characterize now positive solutions of minimal growth in a neighborhoodof infinity of Ω in terms of a modified null sequence of the form a A,V (cf.Section 6).
Theorem 9.4 ([29, Theorem 6.1]) . Suppose that a A,V is nonnegative on C ∞ (Ω) . Let Ω ⋐ Ω be an open set, and let u ∈ C (Ω \ Ω ) be a positivesolution of the equation P u = 0 in Ω \ Ω .Then u ∈ M Ω , Ω if and only if for every smooth open set Ω satisfying Ω ⋐ Ω ⋐ Ω , and an open set B ⋐ (Ω \ Ω ) there exists a sequence { u k } ⊂ C ∞ (Ω) , u k ≥ , such that for all k ∈ N , R B | u k | d x = 1 , and lim k →∞ Z Ω \ Ω u A ∇ u k · ∇ u k d x = 0 . (9.10)Consider now the following Phragm´en–Lindel¨of-type principle that holdsin unbounded or nonsmooth domains, and for irregular potential V , providedthe subsolution satisfies a certain decay property (of variational type) interms of the Lagrangian L (cf. [1, 15, 18, 31] and Section 6). Theorem 9.5 (Comparison Principle [29]) . Assume that P is a nonnegativeSchr¨odinger operator of the form (9.3) . Fix smooth open sets Ω ⋐ Ω ⋐ Ω .Let u, v ∈ W ,p loc (Ω \ Ω ) ∩ C (Ω \ Ω ) be, respectively, a positive subsolutionand a supersolution of the equation P w = 0 in Ω \ Ω such that u ≤ v on ∂ Ω .Assume further that P u ∈ L ∞ loc (Ω \ Ω ) , and that there exist an open set B ⋐ (Ω \ Ω ) and a sequence { u k } ⊂ C ∞ (Ω) , u k ≥ , such that Z B | u k | p d x = 1 ∀ k ≥ , and lim k →∞ Z Ω \ Ω u A ∇ ( u k /u ) · ∇ ( u k /u ) d x = 0 . (9.11) Then u ≤ v on Ω \ Ω . Remark 9.6.
In Theorem 9.5 we assumed that the subsolution u is strictlypositive. It would be useful to prove the above comparison principle underthe assumption that u ≥ cknowledgments Y. P. acknowledges the support of the Israel Science Foundation (grant No.587/07) founded by the Israeli Academy of Sciences and Humanities, by theFund for the Promotion of Research at the Technion, and by the TechnionPresident’s Research Fund.
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