Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
aa r X i v : . [ m a t h . A P ] J a n BILATERAL ESTIMATES OF SOLUTIONSTO QUASILINEAR ELLIPTIC EQUATIONSWITH SUB-NATURAL GROWTH TERMS
I. E. VERBITSKY
Abstract.
We study quasilinear elliptic equations of the type − ∆ p u = σu q + µ in R n in the case 0 < q < p −
1, where µ and σ are nonnegative measurable functions, or locally finite measures,and ∆ p u = div( |∇ u | p − ∇ u ) is the p -Laplacian. Similar equationswith more general local and nonlocal operators in place of ∆ p aretreated as well.We obtain existence criteria and global bilateral pointwise esti-mates for all positive solutions u : u ( x ) ≈ ( W p σ ( x )) p − qp − q − + K p,q σ ( x ) + W p µ ( x ) , x ∈ R n , where W p and K p,q are, respectively, the Wolff potential and theintrinsic Wolff potential, with the constants of equivalence depend-ing only on p , q and n .The contributions of µ and σ in these pointwise estimates aretotally separated, which is a new phenomenon even when p = 2. Inthe homogeneous case µ = 0, such estimates were obtained earlierby a different method only for minimal positive solutions. Contents
1. Introduction 22. Nonlinear potentials 63. Homogeneous equations 74. Non-homogeneous equations 13References 17
Mathematics Subject Classification.
Primary 35J92, 42B37; Secondary35J20.
Key words and phrases.
Wolff potentials, p -Laplacian, pointwise estimates. Introduction
We present a new approach to pointwise estimates of solutions toquasilinear elliptic equations of the type(1.1) ( − ∆ p u = σu q + µ, u ≥ R n , lim inf x →∞ u ( x ) = 0 , where µ , σ ≥ R n , in the sub-natural growth case 0 < q < p − u (possibly unbounded) are understoodto be p -superharmonic (or equivalently locally renormalized) solutions(see [KKT]). We will assume that u ∈ L q loc ( R n , dσ ), so that the right-hand side of (1.1) is a Radon measure.We will obtain matching upper and lower estimates of solutions interms of nonlinear potentials defined below. Our estimates hold forall p -superharmonic solutions u . In particular, they yield an existencecriterion for solutions to (1.1).In the special case µ = 0, i.e.,(1.2) ( − ∆ p u = σu q , u ≥ R n , lim inf x →∞ u ( x ) = 0 , considered earlier in [CV1], the upper pointwise estimate was obtainedonly for the minimal solution u . Our proofs are new even in this case.When p = 2 and 0 < q <
1, these sublinear elliptic equations werestudied by Brezis and Kamin [BK] (see also [CV2], [SV], [QV], [V], andthe literature cited there).The case q ≥ p −
1, which comprises Schr¨odinger type equations with natural growth terms when q = p −
1, and equations of superlinear typewhen q > p −
1, is quite different in nature (see, for instance, [JMV],[JV], [PV1], [PV2]).We observe that in general, for the existence of a nontrivial solution u to (1.1), σ must be absolutely continuous with respect to p -capacity,i.e., σ ( K ) = 0 whenever cap p ( K ) = 0, for any compact set K in R n .Here the p -capacity of K is defined by(1.3) cap p ( K ) = inf (cid:26)Z R n |∇ u | p dx : u ≥ K, u ∈ C ∞ ( R n ) (cid:27) . More precisely, if u is a nontrivial (super)solution to (1.2) in the case0 < q ≤ p −
1, then (see [CV1, Lemma 3.6] for a more general estimate)(1.4) σ ( K ) ≤ cap p ( K ) qp − (cid:18)Z K u q dσ (cid:19) p − − qp − , ILATERAL ESTIMATES OF SOLUTIONS 3 for all compact sets K ⊂ R n .Among our main tools are certain nonlinear potentials associatedwith (1.2). We refer to the recent survey of nonlinear potentials andtheir applications to PDE by Kuusi and Mingione [KuMi].Let M + ( R n ) denote the class of all positive (locally finite) Radonmeasures on R n . Given a measure σ ∈ M + ( R n ), 1 < p < ∞ and0 < α < np , the Havin-Maz’ya-Wolff potential, introduced in [HM] (seealso [HeWo]), is defined by(1.5) W α,p σ ( x ) = Z ∞ (cid:20) σ ( B ( x, t )) t n − αp (cid:21) p − dtt , x ∈ R n , where B ( x, t ) is a ball of radius t > x ∈ R n .Nonlinear potentials W α,p σ , often called Wolff potentials, occur invarious problems of harmonic analysis, approximation theory, Sobolevspaces, in particular spectral synthesis problems ([AH], [HM], [HeWo],[Maz]), as well as quasilinear ([KiMa], [MZ], [PV1]) and fully nonlinearPDE ([Lab], [TW1], [TW2]).In the linear case p = 2, clearly W α,p σ = I α σ (up to a constantmultiple), where the Riesz potential of order β ∈ (0 , n ) is defined by I β σ ( x ) = Z R n dσ ( y ) | x − y | n − β , x ∈ R n . In the special case α = 1, we will be using the notation W p σ = W ,p σ (1 < p < n ), i.e.,(1.6) W p σ ( x ) = Z ∞ (cid:20) σ ( B ( x, t )) t n − p (cid:21) p − dtt , x ∈ R n . These potentials are intimately related to the equation(1.7) ( − ∆ p u = σ, u ≥ R n , lim inf x →∞ u ( x ) = 0 , where σ ∈ M + ( R n ).The following important global estimate, along with its local coun-terpart, is due to T. Kilpel¨ainen and J. Mal´y [KiMa]: Suppose u ≥ is a p -superharmonic solution to (1.7) . Then (1.8) K − W p σ ( x ) ≤ u ( x ) ≤ K W p σ ( x ) , where K = K ( n, p ) is a positive constant. It is known that a nontrivial solution u to (1.7) exists if and only if(1.9) Z ∞ (cid:20) σ ( B (0 , t )) t n − p (cid:21) p − dtt < ∞ . I. E. VERBITSKY
This is equivalent to W p σ ( x ) < ∞ for some x ∈ R n , or equivalentlyquasi-everywhere (q.e.) on R n . In particular, (1.9) may hold only inthe case 1 < p < n , unless σ = 0.The following bilateral pointwise estimates of nontrivial (minimal)solutions u to (1.2) in the case 0 < q < p − c − h ( W p σ ( x )) p − p − − q + K p,q σ ( x ) i ≤ u ( x ) ≤ c h ( W p σ ( x )) p − p − − q + K p,q σ ( x ) i , x ∈ R n , where c > p , q , and n .Here K p,q σ is the so-called intrinsic nonlinear potential associatedwith (1.2), which was introduced in [CV1]. It is defined in terms of thelocalized weighted norm inequalities,(1.11) (cid:18)Z B | ϕ | q dσ (cid:19) q ≤ κ ( B ) || ∆ p ϕ || p − L ( R n ) , for all test functions ϕ such that − ∆ p ϕ ≥
0, lim inf x →∞ ϕ ( x ) = 0. Here κ ( B ) denotes the least constant in (1.11) associated with the measure σ B = σ | B restricted to a ball B = B ( x, t ). Then the intrinsic nonlinearpotential K p,q σ is defined by(1.12) K p,q σ ( x ) = Z ∞ " κ ( B ( x, t )) q ( p − p − − q t n − p p − dtt , x ∈ R n . As was noticed in [CV1], K p,q σ + ∞ if and only if(1.13) Z ∞ " κ ( B (0 , t )) q ( p − p − − q t n − p p − dtt < ∞ . Consequently, a nontrivial p -superharmonic solution u to (1.2) existsif and only if both K p,q σ + ∞ and W p σ + ∞ , that is, both (1.9)and (1.13) hold.For the existence of a nontrivial solution to equation (1.1), we needto add the condition W p µ + ∞ , i.e.,(1.14) Z ∞ (cid:20) µ ( B (0 , t )) t n − p (cid:21) p − dtt < ∞ . In this paper, we obtain the following the following criterion forexistence, along with global bilateral estimates of solutions to (1.1).
ILATERAL ESTIMATES OF SOLUTIONS 5
Theorem 1.1.
Let < p < n , < q < p − , and µ, σ ∈ M + ( R n ) .There exists a nontrivial solution u to (1.1) if and only if conditions (1.9) , (1.13) , and (1.14) hold. Then any nontrivial solution u satisfiesthe estimates (1.15) C h ( W p σ ( x )) p − p − − q + K p,q σ ( x ) + W p µ ( x ) i ≤ u ( x ) ≤ C h ( W p σ ( x )) p − p − − q + K p,q σ ( x ) + W p µ ( x ) i , x ∈ R n . where the positive constants C , C depend only on p, q , and n .If n ≤ p < ∞ , then there are no nontrivial solutions to (1.1) . The following corollary is deduced from Theorem 1.1 under the ad-ditional assumption that there exists a constant C = C ( σ, p, n ) so that(1.16) σ ( K ) ≤ C cap p ( K ) , for all compact sets K ⊂ R n . We remark that condition (1.16) is also essential in the natural growthcase q = p − Corollary 1.2.
Let < p < n , < q < p − , and µ, σ ∈ M + ( R n ) .If condition (1.16) holds, then any positive solution u to (1.1) satisfiesthe estimates (1.17) C h ( W p σ ( x )) p − p − − q + W p µ ( x ) i ≤ u ( x ) ≤ C h ( W p σ ( x )) p − p − − q + W p σ ( x ) + W p µ ( x ) i , x ∈ R n . where C , C are positive constants that depend only on p, q, n , and theconstant C in (1.16) (in the case of C ). In the special case µ = 0, the Brezis–Kamin type pointwise estimates(1.18) C ( W p σ ( x )) p − p − − q ≤ u ( x ) ≤ C [( W p σ ( x )) p − p − − q + W σ ( x )] , under the assumption (1.16) were obtained in [CV2] (the upper esti-mate was proved only for the minimal solution). For bounded solutions u , the term ( W p σ ) p − p − − q on the right-hand side of (1.18) is redundant.This estimate in the case p = 2 was originally obtained in [BK].Our main results are deduced via pointwise estimates of solutions tothe fractional integral equation(1.19) u = W α,p ( u q σ ) + W α,p µ, u ≥ , in R n , where 0 < q < p −
1, 0 < α < np , and µ, σ ∈ M + ( R n ).Bilateral pointwise estimates of solutions to (1.19), similar to (1.10),are given in terms of nonlinear potentials W α,p and fractional intrinsic potentials K α,p,q defined in Sec. 2. In the definition of K α,p,q , whichis similar to (1.12) in the case α = 1, we employ localized embedding I. E. VERBITSKY constants κ ( B ) associated with σ B for balls B = B ( x, r ), which arerelated to certain weighted norm inequalities for potentials W α,p .In the special case p = 2, 0 < q <
1, 0 < α < n , we obtain ananalogue of Theorem 1.1 for the fractional Laplace problem(1.20) ( ( − ∆) α u = σu q + µ, u ≥ R n , lim inf x →∞ u ( x ) = 0 . Our results on solutions to (1.19) demonstrate (see Sec. 4 below)that Theorem 1.1 remains valid for more general quasilinear operatorsdiv A ( x, ∇ u ) in place of ∆ p , under standard boundedness and mono-tonicity assumptions on A ( x, ξ ) (with α = 1, 0 < q < p − k -Hessian operators (with α = kk +1 , p = k + 1 and 0 < q < k ). Therelation between equations (1.19) and the corresponding elliptic PDE isprovided by the nonlinear potential theory developed in [KuMi], [Lab],[TW2].If q ≥ p − q ≥ k for the k -Hessianequations, the existence results and pointwise estimates of solutionsdiffer greatly from Theorem 1.1. They were obtained earlier in [JV],[PV1], [PV2].This paper is organized as follows. In Sec. 2, we recall definitionsof the nonlinear potentials W α,p and K α,p,q , and discuss some of theirproperties. Pointwise estimates of sub- and super-solutions of the ho-mogeneous equation (1.2) are discussed in Sec. 3. They are extendedto the non-homogeneous equation (1.1) in Sec. 4, where we prove The-orem 1.1, and its analogues for equation (1.19).2. Nonlinear potentials
Let 1 < p < ∞ , 0 < α < np , and 0 < q < p −
1. Let σ ∈ M + ( R n ).For the sake of simplicity, the nonlinear potential W α,p σ defined in theIntroduction will be denoted by W σ , i.e.,(2.1) W σ ( x ) = Z ∞ (cid:20) σ ( B ( x, t )) t n − αp (cid:21) p − dtt , x ∈ R n . We denote by κ the least constant in the weighted norm inequality(2.2) || W ν || L q ( R n ,dσ ) ≤ κ ν ( R n ) p − , ∀ ν ∈ M + ( R n ) . We will also need a localized version of (2.2) for σ E = σ | E , where E isa Borel subset of R n , and κ ( E ) is the least constant in(2.3) || W ν || L q ( dσ E ) ≤ κ ( E ) ν ( R n ) p − , ∀ ν ∈ M + ( R n ) . ILATERAL ESTIMATES OF SOLUTIONS 7
In applications, it will be enough to use κ ( E ) where E = Q is a dyadiccube, or E = B is a ball in R n .It is easy to see using estimates (1.8) that embedding constants κ ( B )in the case α = 1 are equivalent to the constants κ ( B ) in (1.11).We define the intrinsic potential of Wolff type K σ = K α,p,q σ in termsof κ ( B ( x, t )), the least constant in (2.3) with E = B ( x, t ):(2.4) K σ ( x ) = Z ∞ " κ ( B ( x, t )) q ( p − p − − q t n − αp p − dtt , x ∈ R n . Notice that K α,p,q σ ( x ) ≈ K p,q σ ( x ) in the case α = 1, with the equiv-alence constants that depend only on p , q , and n (see [CV1]). It is easyto see that K σ ( x )
6≡ ∞ if and only if(2.5) Z ∞ a " κ ( B (0 , t )) q ( p − p − − q t n − αp p − dtt < ∞ , for any (equivalently, all) a >
0. This is similar to the condition W α,p σ ( x )
6≡ ∞ , equivalent to (see, for instance, [CV1, Corollary 3.2])(2.6) Z ∞ a (cid:20) σ ( B (0 , t )) t n − αp (cid:21) p − dtt < ∞ . Homogeneous equations
Let 1 < p < ∞ , 0 < α < np , and 0 < q < p −
1. Let us fix σ ∈ M + ( R n ). We start with some estimates of solutions to the equation(3.1) u ( x ) = W ( u q dσ )( x ) , u ≥ , x ∈ R n , where u < ∞ dσ -a.e. (or equivalently u ∈ L q loc ( R n , σ )). Equation(3.1) can also we considered pointwise at every x ∈ R n where u ( x ) = W ( u q dσ )( x ) < ∞ .We also treat the corresponding subsolutions u ≥ u ( x ) ≤ W ( u q dσ )( x ) < ∞ , x ∈ R n , and supersolutions u ≥ W ( u q dσ ( x ) ≤ u ( x ) < ∞ , x ∈ R n , considered either dσ -a.e., or at every x ∈ R n where these inequalitieshold.For any ν ∈ M + ( R n ) ( ν = 0) such that W ν
6≡ ∞ , we set(3.4) φ ν ( x ) := W ν ( x ) (cid:18) W [( W ν ) q dσ ]( x ) W ν ( x ) (cid:19) p − p − − q , x ∈ R n , I. E. VERBITSKY where we assume that W ν ( x ) < ∞ .Next, for x ∈ R n , we set(3.5) φ ( x ) := sup { φ ν ( x ) : ν ∈ M + ( R n ) , ν = 0 , W ν ( x ) < ∞} . Theorem 3.1.
Let < p < ∞ , < α < np , and < q < p − . Let σ ∈ M + ( R n ) . Then any nontrivial solution u ≥ to (3.1) satisfies theestimates (3.6) C φ ( x ) ≤ u ( x ) ≤ φ ( x ) , x ∈ R n , where C is a positive constant which depends only on p , q , α and n .Moreover, the upper bound in (3.6) holds for any subsolution u ,whereas the lower bound in (3.6) holds for any nontrivial supersolu-tion u .If n ≤ p < ∞ , then there are no nontrivial solutions to (1.1) . The proof of Theorem 3.1 is based on a series of lemmas.
Lemma 3.2.
Let < p < ∞ , < α < np , and < q < p − . Let σ ∈ M + ( R n ) . Suppose u is a subsolution to (3.1) . Then (3.7) u ( x ) ≤ φ ( x ) , x ∈ R n , provided W ( u q dσ )( x ) < ∞ . In paticular, (3.7) holds dσ -a.e.Proof. Setting dν = u q dσ , we see that u ( x ) ≤ W ν ( x ) < ∞ , andconsequently W ν ( x ) ≤ W [( W ν ) q dσ ]( x ). Clearly, φ ν ( x ) := W ν ( x ) (cid:18) W [( W ν ) q dσ ]( x ) W ν ( x ) (cid:19) p − p − − q ≥ W ν ( x ) . Hence, u ( x ) ≤ φ ν ( x ) , x ∈ R n , which yields immediately (3.7). (cid:3) Lemma 3.3.
Let < p < ∞ , < α < np , and < q < p − . Let ν, σ ∈ M + ( R n ) . Then there exists a positive constant C which dependsonly on p , q , α , and n such that (3.8) W [( W ν ) q dσ ]( x ) ≤ C ( W ν ( x )) qp − × h W σ ( x ) + ( K σ ( x )) p − − qp − i , x ∈ R n . Proof.
Without loss of generality we may assume that ν = 0 and W ν ( x ) < ∞ . For x ∈ R n , we have W [( W ν ) q dσ ]( x ) = Z ∞ " R B ( x,t ) ( W ν ( y )) q dσ ( y ) t n − αp p − dtt . (3.9) ILATERAL ESTIMATES OF SOLUTIONS 9
For y ∈ B ( x, t ), we have that B ( y, r ) ⊂ B ( x, t ) if 0 < r ≤ t , and B ( y, r ) ⊂ B ( x, r ) if r > t . Consequently, for y ∈ B ( x, t ), W ν ( y ) = Z t (cid:20) ν ( B ( y, r )) r n − αp (cid:21) p − drr + Z ∞ t (cid:20) ν ( B ( y, r )) r n − αp (cid:21) p − drr ≤ Z t (cid:20) ν ( B ( y, r ) ∩ B ( x, t )) r n − αp (cid:21) p − drr + Z ∞ t (cid:20) ν ( B ( x, r )) r n − αp (cid:21) p − drr ≤ W ν B ( x, t ) ( y ) + c W ν ( x ) , were c = 2 n − αpp − . Hence, Z B ( x,t ) ( W ν ( y )) q dσ ( y ) ≤ Z B ( x,t ) (cid:0) W ν B ( x, t ) (cid:1) q dσ ( y )+ c q ( W ν ( x )) q σ ( B ( x, t )) . Notice that by (2.3), Z B ( x,t ) (cid:0) W ν B ( x, t ) (cid:1) q dσ ( y ) ≤ κ ( B ( x, t )) q ν ( B ( x, t )) qp − . Combining the preceding estimates, we deduce Z B ( x,t ) ( W ν ( y )) q dσ ( y ) ≤ κ ( B ( x, t )) q ν ( B ( x, t )) qp − + c q ( W ν ( x )) q σ ( B ( x, t )) . It follows from (3.9) and the preceding estimate, W [( W ν ) q dσ ]( x ) ≤ c Z ∞ " κ ( B ( x, t )) q ν ( B ( x, t )) qp − t n − αp p − dtt + c ( W ν ( x )) qp − Z ∞ (cid:20) σ ( B ( x, t )) t n − αp (cid:21) p − dtt = c ( I + II ) , where c = c ( p, q, n ). By H¨older’s inequality with exponents p − p − − q and p − q , we estimate I = Z ∞ " κ ( B ( x, t )) q ν ( B ( x, t )) qp − t n − αp p − dtt ≤ Z ∞ (cid:20) ν ( B ( x, t )) t n − αp (cid:21) p − dtt ! qp − × Z ∞ " κ ( B ( x, t ) q ( p − p − − q t n − αp p − dtt p − − qp − ≤ q ( n − αp )( p − ( W ν ( x )) qp − ( K σ ( x )) p − − qp − . Clearly, II = ( W ν ( x )) qp − Z ∞ (cid:20) σ ( B ( x, t )) t n − αp (cid:21) p − dtt = ( W ν ( x )) qp − W σ ( x ) . We deduce W [( W ν ) q dσ ]( x ) ≤ c ( I + II ) ≤ c ( W ν ( x )) qp − h W σ ( x ) + ( K σ ( x )) p − − qp − i . This completes the proof of (3.8). (cid:3)
Lemma 3.4.
Let < p < ∞ , < α < np , and < q < p − . Let σ ∈ M + ( R n ) . Then there exist positive constants C , C which dependonly on p , q , α and n such that (3.10) C φ ( x ) ≤ ( W σ ( x )) p − p − − q + K σ ( x ) ≤ C φ ( x ) , where the lower estimate holds at all x ∈ R n , whereas the upper estimateholds provided W σ ( x ) < ∞ and K σ ( x ) < ∞ . Remark 3.5.
The assumptions W σ ( x ) < ∞ and K σ ( x ) < ∞ inLemma 3.4 can be replaced with W σ
6≡ ∞ and K σ
6≡ ∞ ; then φ ( x ) < ∞ dσ -a.e., and (3.10) holds dσ -a.e. Moreover, the assump-tion W σ ( x ) < ∞ in Lemma 3.4 can be dropped altogether as shownbelow. Proof of Lemma 3.4.
Let ν ∈ M + ( R n ), ν = 0. Suppose W ν ( x ) < ∞ .Raising both sides of (3.8) to the power p − p − − q and multiplying by ILATERAL ESTIMATES OF SOLUTIONS 11 W ν ( x ), we obtain, φ ν ( x ) := W ν ( x ) (cid:18) W [( W ν ) q dσ ]( x ) W ν ( x ) (cid:19) p − p − − q ≤ C p − p − − q h W σ ( x ) + ( K σ ( x )) p − − qp − i p − p − − q . The lower estimate in (3.10) follows immediately from the precedinginequality.To prove the upper estimate in (3.10), notice that, since W σ ( x )
6≡ ∞ and K σ ( x )
6≡ ∞ , it follows by [CV1, Theorem 4.8] that there exists a(minimal) solution u to (3.1) such that(3.11) c h ( W σ ( x )) p − p − − q + K σ ( x ) i ≤ u ( x ) ≤ c h ( W σ ( x )) p − p − − q + K σ ( x ) i , x ∈ R n . Here c , c are positive constants which depend only on p , q , α and n , and (3.11) holds at x provided W σ ( x ) < ∞ and K σ ( x ) < ∞ .Moreover, in this case u ( x ) = W ( u q dσ )( x ) < ∞ . Thus, by Lemma 3.2and the lower bound in (3.11), we have c h ( W σ ( x )) p − p − − q + K σ ( x ) i ≤ u ( x ) ≤ φ ( x ) . The proof of Lemma 3.4 is complete. (cid:3)
Proof of Remark 3.5. If W σ ( x )
6≡ ∞ and K σ
6≡ ∞ , then as indi-cated in the above proof, there exists a solution u to (3.1) such that u = W ( u q dσ ) < ∞ dσ -a.e., and (3.11) holds dσ -a.e. In particular, W σ ( x )
6≡ ∞ and K σ
6≡ ∞ dσ -a.e. Letting dν = u q dσ , we deduce u ≤ φ ν ≤ φ dσ -a.e., so that (3.10) holds dσ -a.e. as well.Let us assume for a moment that W σ ( x ) < ∞ . Then, letting ν = σ in the definition of φ ν , we deduce by [CV1, Lemma 3.5] with r = q , W [( W σ ) q dσ ]( x ) ≥ c ( W σ ( x )) qp − +1 , where c is a positive constant which depends only on p , q , α and n .Hence, φ ( x ) ≥ φ σ ( x ) = W σ ( x ) (cid:18) W [( W σ ) q dσ ]( x ) W σ ( x ) (cid:19) p − p − − q ≥ c ( W σ ( x )) p − p − − q . Next, we observe that in the proof of the upper estimate in (3.10), wemay assume without loss of generality that W σ
6≡ ∞ . Otherwise, wemay replace σ with σ B (0 ,R ) for any R >
0. Then clearly W σ B (0 ,R )
6≡ ∞ , and by the argument presented above (applied to σ B (0 ,R ) in place of σ ),we see that φ ( x ) ≥ c ( W σ B (0 ,R ) ( x )) p − p − − q . Letting R → ∞ and usingthe monotone convergence theorem, we see that the right-hand sidetends to ∞ at every x ∈ R n , which forces φ ≡ ∞ .Finally, if W σ ( x ) = ∞ , but W σ
6≡ ∞ , we may consider W σ k ,where σ k is the p -measure − ∆ p v k = σ k , so that v k ≈ W σ k , with v k =min( v, k ) where − ∆ p v = σ and v ≈ W σ . Notice that W σ k ( x ) = k .Then, clearly, φ σ k ( x ) = k − qp − − q ( W [( W σ k ) q dσ ]( x )) p − p − − q . For k >
0, we set E k = { y : W σ ( y ) ≥ k } , so that W σ E k ( y ) = W σ ( y )for y ∈ E k . We estimate W [( W σ k ) q dσ ]( x ) ≥ k qp − W σ E k ( x ) . Thus, φ ( x ) ≥ φ σ k ( x ) ≥ ( W σ E k ( x )) p − p − − q . Letting k →
0, we see by the monotone convergence theorem that φ ( x ) ≥ φ σ k ( x ) ≥ ( W σ ( x )) p − p − − q = ∞ . In other words, the assumption W σ ( x ) < ∞ in Lemma 3.4 is redun-dant, and actually follows from the fact that φ ( x ) < ∞ . (cid:3) Proof of Theorem 3.1.
The upper bound in (3.6) for any subsolution u follows from Lemma 3.2, whereas the lower bound for any nontrivialsupersolution u is a consequence of Lemma 3.3 and (3.11). (cid:3) As a consequence of the preceding results, we obtain the followingcorollary.
Corollary 3.6.
Under the assumptions of Theorem 3.1, there existpositive constants C , C which depend only on p , q , α and n such that (3.12) C h ( W σ ( x )) p − p − − q + K σ ( x ) i ≤ u ( x ) ≤ C h ( W σ ( x )) p − p − − q + K σ ( x ) i , x ∈ R n , for any solution u to (3.1) . Moreover, the lower estimate holds forany supersolution u such that (3.3) holds at x ∈ R n , whereas the upperestimate holds for any subsolution u such that (3.2) holds at x ∈ R n ,and also dσ -a.e. Remark 3.7.
The upper estimate for u in (3.12) was proved earlier in[CV1] only for the nontrivial minimal solution to (3.1), together withthe lower estimate for any supersolution. ILATERAL ESTIMATES OF SOLUTIONS 13 Non-homogeneous equations
In this section, we deduce estimates for sub- and super-solutions tothe equation(4.1) u = W ( u q dσ ) + W µ, u ≥ R n , in the case 0 < q < p − µ = 0 was considered in Sec. 3, so we assume here that µ = 0.In particluar, all solutions u to (4.1) are nontrivial: u ≥ W µ >
0, and u < ∞ dσ -a.e. (or q.e.) in R n . Theorem 4.1.
Let < p < ∞ , < α < np , and < q < p − .Let σ, µ ∈ M + ( R n ) . Then there exist positive constants C , C whichdepend only on p , q , α and n such that any nonnegative solution u to (4.1) satisfies the estimates (4.2) C h ( W σ ( x )) p − p − − q + K σ ( x ) + W µ ( x ) i ≤ u ( x ) ≤ C h ( W σ ( x )) p − p − − q + K σ ( x ) + W µ ( x ) i , x ∈ R n , where the upper estimate holds at every x where u ( x ) < ∞ , and con-sequently dσ -a.e. and q.e.Moreover, the lower estimate in (4.2) holds for every supersolution u at every x ∈ R n such that (4.3) W ( u q dσ )( x ) + W µ ( x ) ≤ u ( x ) < ∞ , whereas the upper estimate holds for every subsolution u at every x ∈ R n such that (4.4) u ( x ) ≤ W ( u q dσ )( x ) + W µ ( x ) < ∞ . Proof.
The case µ = 0 is considered in Sec. 3, so we may assume with-out loss of generality that µ = 0. Consequently, u ( x ) ≥ W µ ( x ) > x ∈ R n . Clearly, any supersolution to (4.1) is also a su-persolution to (3.1). Hence, by Theorem 3.1, there exists a positiveconstant c = c ( p, q, α, n ) such that u ( x ) ≥ c h ( W σ ( x )) p − p − − q + K σ ( x ) i .These two lower estimates combined yield the lower bound in (4.2) with C = C ( p, q, α, n ) > u to (4.2), we fix x ∈ R n such that u ( x ) ≤ W ( u q dσ )( x ) + W µ ( x ) < ∞ . Notice that if u is a solution to (4.2), then this is equivalent to u ( x ) < ∞ .Letting dω = u q dσ + dµ and c = max (cid:16) , p − p − (cid:17) , we obviously have u ( x ) ≤ c W ω ( x ) < ∞ at x and dσ -a.e. Letting c = max (cid:16) , − p (cid:17) , we estimate W ω ( x ) = W ( u q dσ + dµ )( x ) ≤ c W ( u q dσ )( x ) + c W µ ( x ) ≤ c q c W [( W ω ) q dσ ]( x ) + c W µ ( x ) . By Lemma 3.3 with ω in place of ν , we have W [( W ω ) q dσ ]( x ) ≤ C ( W ω ( x )) qp − h ( W σ ( x )) p − p − − q + K σ ( x ) i p − − qp − , where C = C ( p, q, α, n ) is a positive constant. Combining the precedingestimates we deduce W ω ( x ) ≤ c q c C ( W ω ( x )) qp − h ( W σ ( x )) p − p − − q + K σ ( x ) i p − − qp − + c W µ ( x ) . Using Young’s inequality with exponents p − q and p − p − − q in the firstterm on the right-hand side, we estimate W ω ( x ) ≤ W ω ( x ) + C ′ h ( W σ ( x )) p − p − − q + K σ ( x ) i + c W µ ( x ) , where C ′ is a positive constant which depends only on p , q , α and n .Since W ω ( x ) < ∞ , we can move the first term on the right to theleft-hand side, and obtain u ( x ) ≤ c W ω ( x ) ≤ C h ( W σ ( x )) p − p − − q + K σ ( x ) + W µ ( x ) i , where C is a positive constant which depends only on p , q , α and n .This completes the proof of the upper estimate in (4.2). (cid:3) Proof of Theorem 1.1.
Let dω = u q dσ + dµ , where u is a solution to(1.1). Then by (1.8),(4.5) K − W p ω ( x ) ≤ u ( x ) ≤ K W p ω ( x ) , where K = K ( n, p ) is a positive constant. Hence, u is a supersolutionsatisfying u ≥ W p ( u q d ˜ σ ) + W p ˜ µ , with ˜ µ = c µ and ˜ σ = c σ , where c , c depend only on p , q , and K . Hence the lower estimate (1.15)of Theorem 1.1 follows from the lower estimate (4.2) of Theorem 4.1in the special case α = 1. Similarly, the upper estimate in (1.15) isdeduced from the upper estimates in (4.2) and (4.5).If a nontrivial solution u to (1.1) exists, then by the lower estimate(1.15) of Theorem 1.1 it follows that W p µ
6≡ ∞ , W p σ
6≡ ∞ , and K p σ
6≡ ∞ , which are equivalent to conditions (1.14), (1.9), and (1.13),respectively.
ILATERAL ESTIMATES OF SOLUTIONS 15
Conversely, suppose that these three conditions hold. In the specialcase µ = 0, a positive p -superharmonic solution u ∈ L q loc ( R n ) wasconstructed in [CV1, Theorem 1.1] by iterations, u = lim j →∞ u j , where u j is an nondecreasing sequence of p -superharmonic functions such that(4.6) − ∆ p u j +1 = σu qj + µ in R n , j = 0 , , , . . . , with an appropriate choice of u , namely u = c ( W p σ ) p − p − − q , where c = c ( p, q, n ) is a small constant.If µ = 0, a similar iteration argument can be used with u = 0 basedon [PV2, Lemma 3.7 and Lemma 3.9], so that u j satisfying (4.6) is annondecreasing sequence of p -superharmonic functions. This part of theconstruction works for any q > p > q > p − u j for 0 < q < p − u j ≤ u j +1 , it follows that u j +1 is a subsolution, so that(4.7) − ∆ p u j +1 ≤ σu qj +1 + µ in R n , j = 0 , , , . . . , By (1.8), we have(4.8) u j +1 ≤ K W p ( σu qj + µ ) ≤ K max(1 , − pp − ) (cid:2) W p ( σu qj +1 ) + W p µ (cid:3) . After scaling by letting ˜ µ = c p − µ and ˜ σ = c p − σ , where the constant c = K max(1 , − pp − ), we see that u j +1 is a subsolution for the corre-sponding integral equation (4.1), i.e.,(4.9) u j +1 ≤ W p (˜ σu qj +1 ) + W p ˜ µ, j = 0 , , , . . . . It follows by induction using Lemma 3.3 with ν = ˜ µ and ν = ˜ σu qj that the right-hand side of (4.9) is finite at every point x ∈ R n where W p µ ( x ) < ∞ , W p σ ( x ) < ∞ , and K p σ ( x ) < ∞ (which is true dσ -a.e.,as we demonstrate below).By Theorem 4.1 for subsolutions, u j +1 has the upper bound u j +1 ( x ) ≤ C h ( W p σ ( x )) p − p − − q + K p σ ( x ) + W p µ ( x ) i , x ∈ R n , with C that depends only on p , q , and n , where we switched back from˜ µ , ˜ σ to µ , σ .Thus, u = lim j →∞ u j satisfies(4.10) u ( x ) ≤ C h ( W p σ ( x )) p − p − − q + K p σ ( x ) + W p µ ( x ) i , x ∈ R n . Moreover, by [CV1, Theorem 1.1], the conditions W p σ
6≡ ∞ and K p σ
6≡ ∞ yield the existence of a positive solution v ∈ L q loc ( R n , σ ) to the homogeneous equation − ∆ p v = σv q in R n , so that v satisfies the lower bound v ≥ c h ( W p σ ) p − p − − q + K p σ i , where c > p , q , and n . Hence,( W p σ ) p − p − − q ∈ L q loc ( R n , σ ) and K p σ ∈ L q loc ( R n , σ ).To verify that u ∈ L q loc ( R n , σ ), in view of (4.10), it remains to showthat W p µ ∈ L q loc ( R n , σ ). Let B = B (0 , R ) and let µ = µ B + µ (2 B ) c .Then, as in the proof of Lemma 3.3, we clearly have for all x ∈ B , W p µ (2 B ) c ( x ) = Z ∞ (cid:20) µ ( B ( x, t ) ∩ (2 B ) c ) t n − p (cid:21) p − dtt ≤ Z ∞ R (cid:20) µ ( B (0 , t ) t n − p (cid:21) p − dtt = 2 n − pp − Z ∞ R (cid:20) µ ( B (0 , t ) t n − p (cid:21) p − dtt . Hence, Z B ( W p µ ) q dσ ≤ c Z B ( W p µ B ) q dσ + c Z B ( W p µ (2 B ) c ) q dσ ≤ c κ ( B ) q µ (2 B ) qp − + c σ ( B ) Z ∞ R (cid:20) µ ( B (0 , t )) t n − αp (cid:21) p − dtt ! q , where c > p , q , and n . The right-hand side of the preceding estimate is obviously finite by (1.13) and(1.14). This proves that u ∈ L q loc ( R n , σ ).Passing to the limit as j → ∞ in (4.6), we deduce as in [PV1], [PV2]for q > p − u is a positive p -superharmonic solution to (1.1). (cid:3) Corollary 4.2.
The results involving pointwise estimates, as well asnecessary and sufficient conditions for u ∈ W ,p loc ( R n ) , u ∈ L r ( R n ) , u ∈ L r loc ( R n ) , etc., obtained in [CV1] , [CV2] , [SV] , [V] for minimalsolutions u , actually hold for all solutions. Remark 4.3.
1. In the case p = 2, Theorem 4.1 yields bilateral point-wise estimates of solutions to the fractional Laplace equation (1.20).The case of homogeneous equations µ = 0 was considered earlier in[CV1], where the upper estimate was proved only for the minimal so-lution u . ILATERAL ESTIMATES OF SOLUTIONS 17
2. As was mentioned in the Introduction, Theorem 1.1 is valid forgeneral quasilinear A -Laplace operators div A ( x, ∇ u ) in place of ∆ p under the standard structural assumptions on A which ensure that theKilpel¨ainen–Mal´y estimates (1.8) hold (see [KiMa], [MZ]). The proofsin this setup are identical to those given above, with the same nonlinearpotentials W p and K p,q . The constants in our pointwise estimates(1.15) then depend on the structural constants of A . Analogous resultsalso hold for k -Hessian equations in the case 0 < q < k (see [CV1] for µ = 0, and [PV2] for q > k ).3. Complete analogues of our results for (1.1) hold for the non-homogeneous problem ( − ∆ p u = σu q + µ, u ≥ R n , lim inf x →∞ u ( x ) = c, where c is a positive constant. One only needs to add c to both sidesof (1.15). References [AH]
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Department of Mathematics, University of Missouri, Columbia,Missouri 65211, USA
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