Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms
aa r X i v : . [ m a t h . A P ] S e p FINITE ENERGY SOLUTIONS OFQUASILINEAR ELLIPTIC EQUATIONSWITH SUB-NATURAL GROWTH TERMS
CAO TIEN DAT AND IGOR E. VERBITSKY
Abstract.
We study finite energy solutions to quasilinear elliptic equa-tions of the type − ∆ p u = σ u q in R n , where ∆ p is the p -Laplacian, p >
1, and σ is a nonnegative function (ormeasure) on R n , in the case 0 < q < p − q = p − σ which ensures that there exists a solution u in the homogeneousSobolev space L ,p ( R n ), and prove its uniqueness. Among our maintools are integral inequalities closely associated with this problem, andWolff potential estimates used to obtain sharp bounds of solutions. Moregeneral quasilinear equations with the A -Laplacian div A ( x, ∇· ) in placeof ∆ p are considered as well. Introduction
This paper is concerned with quasilinear problems of the following type:(1.1) − ∆ p u = σ u q in R n , where ∆ p u = ∇ · ( ∇ u |∇ u | p − ) is the p -Laplacian, 1 < p < ∞ , and σ is anonnegative function, or measure, in the sub-natural growth case 0 < q
1. We are interested in finite energy solutions u ∈ L ,p ( R n ) to (1.1),and related integral inequalities. Here L ,p ( R n ) is the homogeneous Sobolev(or Dirichlet) space defined in Sec. 2 (see [HKM06], [MZ97], [Maz11]); for1 < p < n it can be identified with the completion of C ∞ ( R n ) in the norm(1.2) || u || ,p = (cid:16) Z R n |∇ u ( x ) | p dx (cid:17) p . More precisely, u is called a finite energy solution to (1.1) if u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ), u ≥
0, and, for all ϕ ∈ C ∞ ( R n ),(1.3) Z R n |∇ u | p − ∇ u · ∇ ϕ dx = Z R n u q ϕ dσ. Key words and phrases.
Quasilinear equations, finite energy solutions, p -Laplacian,Wolff’s potentials.Supported in part by NSF grant DMS-1161622. Finite energy solutions to (1.1) are critical points of the functional H [ ϕ ] = Z R n p |∇ ϕ | p dx − Z R n q + 1 | ϕ | q dσ. We will give a necessary and sufficient condition for the existence of afinite energy solution to (1.1), and prove its uniqueness.Our results are new even in the classical case p = 2, 0 < q <
1. Sublinearelliptic problems of this type were studied by Brezis and Kamin in [BrK92],where a necessary and sufficient condition is found for the existence of a bounded solution on R n , together with sharp pointwise estimates of solutions.Recently, we have extended these results to the case p = 2, under relaxedassumptions on σ , in such a way that some singular (unbounded) solutionsare covered as well [CV13]. However, the techniques used in [CV13] arequite different from those used in this paper.Analogous sublinear problems in bounded domains Ω ⊂ R n for variousclasses of σ have been extensively studied. In particular, Boccardo andOrsina [BO96], [BO12], and Abdel Hamid and Bidaut-V´eron [ABV10] gavesufficient conditions for the existence of solutions under the assumption σ ∈ L r (Ω). Earlier results, under more restrictive assumptions on σ , can befound in Krasnoselskii [Kr64], Brezis and Oswald [BrO86], and the literaturecited in these papers.We employ powerful Wolff potential estimates developed in [KM94] (seealso [Lab02], [TW02], [KuMi13]). This makes it possible to replace the p -Laplacian ∆ p in the model problem (1.1) by a more general quasilin-ear operator div A ( x, ∇· ) with bounded measurable coefficients, under stan-dard structural assumptions on A ( x, ξ ) which ensure that A ( x, ξ ) · ξ ≈ | ξ | p [HKM06], [MZ97], or a fully nonlinear operator of k -Hessian type [TW99],[Lab02] (see also [PV09], [JV12]), and treat more general nonlinearities onthe right-hand side. Equations involving operators of the p -Laplacian typeon Carnot groups can be covered as well using methods developed in [PV13].Wolff’s potential W ,p σ of a nonnegative Borel measure σ on R n is definedby [HW83] (see also [AH96]):(1.4) W ,p σ ( x ) = Z ∞ (cid:18) σ ( B ( x, t )) t n − p (cid:19) p − dtt . Here B ( x, t ) = { y ∈ R n : | x − y | < t } is a ball centered at x ∈ R n of radius t > U is a solution (understood in the potential theoreticor renormalized sense) to the equation(1.5) ( − ∆ p U = σ in R n , inf R n U = 0 , INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 3 then there exists a constant
K > p and n such that(1.6) 1 K W ,p σ ( x ) ≤ U ( x ) ≤ K W ,p σ ( x ) , x ∈ R n . Moreover, U exists if and only if W ,p σ + ∞ (see [PV08]), or equivalently,(1.7) Z ∞ (cid:18) σ ( B (0 , t )) t n − p (cid:19) p − dtt < + ∞ . Our main result is the following
Theorem.
Let < q < p − , < p < n , and let σ be a locally finite positivemeasure on R n . Then there exists a nontrivial solution u ∈ L ,p ( R n ) ∩ L q loc (Ω , dσ ) to (1.1) if and only if U ∈ L (1+ q )( p − p − − q ( R n , dσ ) , or equivalently, (1.8) Z R n ( W ,p σ ) (1+ q )( p − p − − q dσ < ∞ . Furthermore, such a solution is unique. For p ≥ n , (1.1) has only a trivialsolution u = 0 . We observe that (1.8) yields σ ∈ L − ,p ′ loc ( R n ), where L − ,p ′ ( R n ) = L ,p ( R n ) ∗ is the dual Sobolev space (see definitions in Sec. 2). Consequently, σ is nec-essarily absolutely continuous with respect to the p -capacity cap p ( · ) definedby(1.9) cap p ( E ) = inf {||∇ φ || pL p : φ ≥ E , φ ∈ C ∞ ( R n ) } , for a compact set E ⊂ R n .Moreover, as was shown in [COV00], condition (1.8) holds if and only ifthere exists a constant C such that, for all ϕ ∈ C ∞ ( R n ),(1.10) (cid:16) Z R n | ϕ | q dσ (cid:17) q ≤ C ||∇ ϕ || L p ( R n ) . An obvious sufficient condition which follows from Sobolev’s inequality is σ ∈ L r ( R n ), r = npn ( p − − q )+ p (1+ q ) .There is also an equivalent characterization of (1.10) in terms of capacitiesdue to Maz’ya and Netrusov (see [Maz11], Sec. 11.6):(1.11) Z σ ( R n )0 (cid:20) t κ ( σ, t ) (cid:21) qp − − q dt < + ∞ , where κ ( σ, t ) = inf { cap p ( E ) : σ ( E ) ≥ t } .Thus, any one of the conditions (1.8), (1.10), and (1.11) is necessary andsufficient for the existence of a nontrivial finite energy solution to (1.1).We now outline the contents of the paper. Sec. 2 contains definitions andnotations, along with several useful results on quasilinear equations that willbe used below. In Sec. 3 we study the corresponding integral inequalities,deduce a necessary and sufficient condition for the existence of a finite energysolution, and construct a minimal solution. Sec. 4 is devoted to more general CAO TIEN DAT AND IGOR E. VERBITSKY equations with the operator div A ( x, ∇· ) in place of the p -Laplacian. In Sec. 5we prove the uniqueness property of finite energy solutions.2. Preliminaries
We first recall some notations and definitions. Given an open set Ω ⊆ R n ,we denote by M + (Ω) the class of all nonnegative Borel measures in Ω whichare finite on compact subsets of Ω. The σ -measure of a measurable set E ⊂ Ω is denoted by | E | σ = σ ( E ) = R E dσ. For p > σ ∈ M + (Ω), we denote by L p (Ω , dσ ) ( L p loc (Ω , dσ ), re-spectively) the space of measurable functions ϕ such that | ϕ | p is integrable(locally integrable) with respect to σ . For u ∈ L p (Ω , dσ ), we set || u || L p (Ω ,dσ ) = (cid:16) Z Ω | u | p dσ (cid:17) p . When dσ = dx , we write L p (Ω) (respectively L p loc (Ω)), and denote Lebesguemeasure of E ⊂ R n by | E | .The Sobolev space W ,p (Ω) ( W ,p loc (Ω), respectively) is the space of allfunctions u such that u ∈ L p (Ω) and |∇ u | ∈ L p (Ω) ( u ∈ L p loc (Ω) and |∇ u | ∈ L p loc (Ω), respectively). By L ,p (Ω) we denote the homogeneous Sobolevspace, i.e., the space of functions u ∈ W ,p loc (Ω) such that |∇ u | ∈ L p (Ω),and ||∇ u − ∇ ϕ j || L p (Ω) → j → ∞ for a sequence ϕ j ∈ C ∞ (Ω).When 1 < p < n and Ω = R n , we will identify L ,p ( R n ) with the spaceof all functions u ∈ W ,p loc ( R n ) such that u ∈ L npn − p ( R n ) and |∇ u | ∈ L p ( R n ).For u ∈ L ,p ( R n ), the norm || u || ,p is equivalent to || u || L npn − p ( R n ) + ||∇ u || L p ( R n ) . It is easy to see that C ∞ ( R n ) is dense in L ,p ( R n ) with respect to this norm(see, e.g., [MZ97], Sec. 1.3.4).If 1 < p < n and Ω = R n , then the dual Sobolev space L − ,p ′ ( R n ) = L ,p ( R n ) ∗ is the space of distributions ν such that || ν || − ,p ′ = sup |h u, ν i||| u || ,p < + ∞ , where the supremum is taken over all u ∈ L ,p ( R n ), u = 0. We write ν ∈ L − ,p ′ loc ( R n ) if ϕ ν ∈ L − ,p ′ ( R n ), for every ϕ ∈ C ∞ ( R n ).For u ∈ W ,p loc (Ω), we define the p -Laplacian ∆ p (1 < p < ∞ ), in thedistributional sense, i.e., for every ϕ ∈ C ∞ (Ω),(2.1) h ∆ p u, ϕ i = h div( |∇ u | p − ∇ u, ϕ i = − Z Ω |∇ u | p − ∇ u · ∇ ϕ dx. INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 5 A finite energy solution u ≥ u ∈ L ,p (Ω) ∩ L q loc (Ω , dσ ), and, for every ϕ ∈ C ∞ (Ω),(2.2) Z Ω |∇ u | p − ∇ u · ∇ ϕ dx = Z Ω u q ϕ dσ. We need to extend the definition of solutions to u not necessarily in W ,p loc (Ω). We will understand solutions in the following potential-theoreticsense using p -superharmonic functions, which is equivalent to the notion oflocally renormalized solutions in terms of test functions (see [KKT09]).A function u ∈ W ,p loc (Ω) is called p -harmonic if it satisfies the homo-geneous equation ∆ p u = 0. Every p -harmonic function has a continuousrepresentative which coincides with u a.e. (see [HKM06]).As usual, p -superharmonic functions are defined via a comparison prin-ciple. We say that u : Ω → ( −∞ , ∞ ] is p -superharmonic if u is lower semi-continuous, is not identically infinite in any component of Ω, and satisfiesthe following comparison principle: Whenever D ⊂⊂ Ω and h ∈ C ( ¯ D ) is p -harmonic in D , with h ≤ u on ∂D , then h ≤ u in D .A p -superharmonic function u does not necessarily belong to W ,p loc (Ω),but its truncates T k ( u ) = min( u, k ) do, for all k >
0. In addition, T k ( u )are supersolutions, i.e., − div( |∇ T k ( u ) | p − ∇ T k ( u )) ≥
0, in the distributionalsense. We will need the generalized gradient of a p -superharmonic function u defined by [HKM06]: Du = lim k →∞ ∇ ( T k ( u )) . We note that every p -superharmonic function u has a quasicontinuous rep-resentative which coincides with u quasieverywhere (q.e.), i.e., everywhereexcept for a set of p -capacity zero. We will assume that u is always chosenthis way.Let u be p -superharmonic, and let 1 ≤ r < nn − . Then | Du | p − , andhence | Du | p − Du , belong to L r loc (Ω) [KM92]. This allows us to define anonnegative distribution − ∆ p u for each p -superharmonic function u by(2.3) − h ∆ p u, ϕ i = Z Ω | Du | p − Du · ∇ ϕ dx, for all ϕ ∈ C ∞ (Ω). Then by the Riesz representation theorem there existsa unique measure µ [ u ] ∈ M + (Ω) so that − ∆ p u = µ [ u ]. Definition 2.1.
For a nonnegative locally finite measure ω in Ω we will saythat − ∆ p u = ω in Ω in the potential-theoretic sense if u is p -superharmonic in Ω , and µ [ u ] = ω .Thus, − ∆ p u = σu q if u ≥ is p -superharmonic in Ω , u ∈ L qloc (Ω , dσ ) ,and dµ [ u ] = u q dσ . CAO TIEN DAT AND IGOR E. VERBITSKY
Definition 2.2.
A function u ≥ is a supersolution to (1.1) if u is p -superharmonic, u ∈ L qloc (Ω , dσ ) , and, for every nonnegative ϕ ∈ C ∞ (Ω) , (2.4) Z Ω | Du | p − Du · ∇ ϕ dx ≥ Z Ω u q ϕ dσ. Supersolutions to (1.1) in the sense of Definition 2.2 are closely related tosupersolutions associated with the integral equation(2.5) u = W ,p ( u q dσ ) dσ -a . e ., that is, measurable functions u ≥ W ,p ( u q dσ ) ≤ u < ∞ dσ -a . e . We will use the following universal lower bound for supersolutions ob-tained in [CV13].
Theorem 2.3.
Let < p < n , < q < p − , and σ ∈ M + ( R n ) . Suppose u is a nontrivial p -superharmonic supersolution to (1.1) . Then the inequality (2.6) u ≥ C (cid:0) W ,p σ (cid:1) p − p − − q holds, where C is a positive constant depending only on p, q , and n .The same lower bound holds for a nontrivial supersolution to the integralequation (2.5) . If p ≥ n , there is only a trivial supersolution u = 0 on R n . We will employ some fundamental results of the potential theory of quasi-linear elliptic equations. The following important weak continuity result[TW02] will be used to prove the existence of p -superharmonic solutions toquasilinear equations. Theorem 2.4.
Suppose { u n } is a sequence of nonnegative p -superharmonicfunctions that converges a.e. to a p -superharmonic function u in an openset Ω . Then µ [ u n ] converges weakly to µ [ u ] , i.e., for all ϕ ∈ C ∞ (Ω) , lim n →∞ Z Ω ϕ dµ [ u n ] = Z Ω ϕ dµ [ u ] . The next result [KM94] is concerned with global pointwise estimates ofnonnegative p -superharmonic functions in terms of Wolff’s potentials dis-cussed in the Introduction. Theorem 2.5.
Let < p ≤ n . Let u be a p -superharmonic function in R n with inf R n u = 0 . If ω is a nonnegative Borel measure in R n such that − ∆ p u = ω , then K W ,p ω ( x ) ≤ u ( x ) ≤ K W ,p ω ( x ) , x ∈ R n , where K is a positive constant depending only on n, p. The following theorem is due to Brezis and Browder [BrB79] (see also[MZ97], Theorem 2.39).
INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 7
Theorem 2.6.
Let < p < n . Suppose u ∈ L ,p ( R n ) , and µ ∈ M + ( R n ) ∩ L − ,p ′ ( R n ) . Then u ∈ L ( R n , µ ) (for a quasicontinuous representative of u ),and (2.7) h µ, u i = Z R n u dµ. We observe that if, under the assumptions of this theorem, − ∆ p u = µ ,then it follows (see [MZ97], Theorem 2.34)(2.8) h µ, u i = Z R n u dµ = || u || p ,p = || µ || p ′ − ,p ′ . For 0 < α < n and σ ∈ M + ( R n ), the Riesz potential of σ is defined by(2.9) I α σ ( x ) = Z ∞ σ ( B ( x, r )) r n − α drr = 1 n − α Z R n dσ ( y ) | x − y | n − α , x ∈ R n . For 1 < p < ∞ and 0 < α < np , the Wolff potential of order α is definedby W α,p σ ( x ) = Z ∞ (cid:18) σ ( B ( x, s )) s n − αp (cid:19) p − dss , x ∈ R n . Note that W α, σ = I α σ if 0 < α < n . In particular, W , σ = I σ is theNewtonian potential for n ≥ Theorem 2.7.
Suppose < p < ∞ , < α < np , and σ ∈ M + ( R n ) . Thenthere exists a constant C > depending only on p, α , and n such that (2.10) 1 C Z R n ( I α σ ) p ′ dx ≤ Z R n W α,p σ dσ ≤ C Z R n ( I α σ ) p ′ dx, where p + p ′ = 1 . Existence and minimality of finite energy solutions
In this section, we deduce a necessary and sufficient condition for theexistence of a finite energy solution, and construct a minimal solution to(1.1). We will assume that 1 < p < n , since for p ≥ n there are onlytrivial nonnegative supersolutions on R n (Theorem 2.3; see also [HKM06],Theorem 3.53). Lemma 3.1.
Suppose there exists a nontrivial supersolution u ≥ , u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) to (1.1) . Then − ∆ p u ∈ L − ,p ′ ( R n ) ∩ M + ( R n ) . Moreover, u ∈ L q ( R n , σ ) (for a quasicontinuous representative of u ), andcondition (1.8) holds. CAO TIEN DAT AND IGOR E. VERBITSKY
Proof.
Suppose u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) is a supersolution to (1.1). Thenby H¨older’s inequality, for every ϕ ∈ C ∞ ( R n ), |h ∆ p u, ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z R n |∇ u | p − ∇ u · ∇ ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ||∇ u || p − L p ( R n ) ||∇ ϕ || L p ( R n ) . Hence, ∆ p u ∈ L − ,p ′ ( R n ). If ϕ ≥
0, then −h ∆ p u, ϕ i = Z R n |∇ u | p − ∇ u · ∇ ϕ dx ≥ Z R n ϕ u q dσ ≥ , and consequently − ∆ p u ∈ M + ( R n ).It follows that dµ = u q dσ ∈ M + ( R n ) ∩ L − ,p ′ ( R n ). Let { ϕ j } be a sequenceof nonnegative C ∞ -functions such that ϕ j → u in L ,p ( R n ). By definition, Z R n |∇ u | p − ∇ u · ∇ ϕ j dx ≥ h µ, ϕ j i . Hence, Z R n |∇ u | p dx = lim j →∞ Z R n |∇ u | p − ∇ u · ∇ ϕ j dx ≥ lim j →∞ h µ, ϕ j i = h µ, u i . Let us assume as usual that u coincides with its quasicontinuous represen-tative. Then, applying Theorem 2.6, we deduce h µ, u i = Z R n u dµ = Z R n u q dσ < ∞ . By Theorem 2.3, it follows that if u
0, then u ≥ C (cid:0) W ,p σ (cid:1) p − p − − q , andconsequently (1.8) holds. (cid:3) Lemma 3.2.
For every r > , (3.1) W α,p (( W α,p σ ) r dσ )( x ) ≥ C ( W α,p σ ( x )) rp − +1 , x ∈ R n , where C depends only on p , q , r , α , and n .Proof. For t >
0, obviously, W α,p σ ( y ) = Z t (cid:18) σ ( B ( y, s )) s n − αp (cid:19) p − dss + Z ∞ t (cid:18) σ ( B ( y, s )) s n − αp (cid:19) p − dss For y ∈ B ( x, t ), we have Z ∞ t (cid:18) σ ( B ( y, s )) s n − αp (cid:19) p − dss = Z ∞ t/ (cid:18) σ ( B ( y, r ))(2 r ) n − αp (cid:19) p − drr = (cid:18) (cid:19) n − αpp − Z ∞ t/ (cid:18) σ ( B ( y, s )) s n − αp (cid:19) p − dss ≥ C n,p,α Z ∞ t (cid:18) σ ( B ( y, s )) s n − αp (cid:19) p − dss , INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 9 where C n,p,α = (cid:0) (cid:1) n − αpp − . Since s ≥ t and y ∈ B ( x, t ), then B ( y, s ) ⊃ B ( x, s ), which implies(3.2) W α,p σ ( y ) ≥ C n,p,α Z ∞ t (cid:18) σ ( B ( x, s )) s n − αp (cid:19) p − dss . Notice that W α,p (( W α,p σ ) r dσ )( x ) = Z ∞ R B ( x,t ) [ W α,p σ ( y )] r dσ ( y ) t n − αp ! p − dtt . By (3.2), we obtain W α,p (( W α,p σ ) r dσ )( x ) ≥≥ Z ∞ R B ( x,t ) h C n,p,α R ∞ t (cid:16) σ ( B ( x,s )) s n − αp (cid:17) p − dss i r dσ ( y ) t n − αp p − dtt ≥ C rp − n,p,α Z ∞ hZ ∞ t (cid:18) σ ( B ( x, s )) s n − αp (cid:19) p − dss i rp − (cid:18) σ ( B ( x, t )) t n − αp (cid:19) p − dtt . Integrating by parts, we deduce W α,p (( W α,p σ ) r dσ )( x ) ≥ C rp − n,p,αrp − + 1 Z ∞ (cid:18) σ ( B ( x, s )) s n − αp (cid:19) p − dss ! rp − +1 . Thus, W α,p (( W α,p σ ) r dσ )( x ) ≥ C n,p,α,r ( W ,p σ ( x )) rp − +1 . (cid:3) Setting r = q ( p − p − − q in Lemma 3.2, we deduce(3.3) W α,p (( W α,p σ ) q ( p − p − − q dσ )( x ) ≥ κ ( W α,p σ ( x )) p − p − − q , where κ depends only on p , q , and n .Let us define a nonlinear integral operator T by(3.4) T ( f )( x ) = (cid:16) W α,p ( f dσ ) (cid:17) p − ( x ) , x ∈ R n . Lemma 3.3.
Let < p < ∞ , < α < n , and < q < p − . Suppose (3.5) Z R n ( W α,p σ ) (1+ q )( p − p − − q dσ < ∞ . Then T is a bounded operator from L qq ( R n , dσ ) to L qp − ( R n , dσ ) . Proof.
Clearly, || ( W α,p ( f dσ )) p − || L qp − ( dσ ) = (cid:18)Z R n (cid:16) W α,p ( f dσ ) (cid:17) q dσ (cid:19) p − q . We have W α,p ( f dσ )( x ) ≤ Z ∞ (cid:18) σ ( B ( x, r )) r n − αp (cid:19) p ′ − M σ f ( x ) p ′ − drr = M σ f ( x ) p ′ − W ,p σ ( x ) , where the centered maximal operator M σ is defined by M σ f ( x ) = sup r> σ ( B ( x, r )) Z B ( x,r ) | f | dσ, x ∈ R n . It is well known that M σ : L s ( R n , dσ ) → L s ( R n , dσ ) is a bounded operatorfor all s >
1. Let s = qq . Then, using H¨older’s inequality with theexponents β = p − q > β ′ = p − p − − q , we estimate, Z R n (cid:16) W α,p ( f dσ ) (cid:17) q dσ ≤ Z R n ( M σ f ) qp − ( W α,p σ ) q dσ ≤ (cid:18)Z R n ( M σ f ) qq dσ (cid:19) qp − (cid:18)Z R n ( W α,p σ ) (1+ q )( p − p − − q dσ (cid:19) p − − qp − ≤ C (cid:18)Z R n f qq dσ (cid:19) qp − (cid:18)Z R n ( W α,p σ ) (1+ q )( p − p − − q dσ (cid:19) p − − qp − . Thus, || W α,p ( f dσ ) p − || L qp − ( dσ ) ≤ c || f || L qq ( dσ ) . (cid:3) Remark 3.4.
It is not difficult to see that actually (3.5) is also necessaryfor the boundedness of the operator T : L qq ( R n , dσ ) → L qp − ( R n , dσ ) (see,for example, [COV06] ). Theorem 3.5.
Let < p < n , and < q < p − . Suppose that condition (1.8) holds. Then there exists a solution u ∈ L q ( R n , dσ ) to the integralequation (2.5) .Proof. By Lemma 3.3, we have, for all f ∈ L qq ( R n , dσ ),(3.6) Z R n (cid:16) W ,p ( f dσ ) (cid:17) q dσ ≤ C (cid:18)Z R n f qq dσ (cid:19) qp − . Let u = c ( W ,p σ ) p − p − − q , where c > u j as follows:(3.7) u j +1 = W ,p ( u qj dσ ) , j = 0 , , , . . . . Applying Lemma 3.2, we have u = W ,p ( u q dσ ) = c qp − W ,p (( W ,p σ ) q ( p − p − − q dσ ) ≥ c qp − κ ( W ,p σ ) p − p − − q , INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 11 where κ is the constant in (3.1). Choosing c so that c qp − κ ≥ c , we obtain u ≥ u . By induction, we can show that the sequence { u j } is nondecreasing.Note that u ∈ L q ( R n , dσ ) by assumption. Suppose that u , . . . , u j ∈ L q ( R n , dσ ). Then Z R n u qj +1 dσ = Z R n ( W ,p ( u qj dσ ) q dσ. Applying (3.6) with f = u qj , we obtain by induction,(3.8) Z R n u qj +1 dσ ≤ C (cid:18)Z R n u qj dσ (cid:19) qp − < ∞ . Since u j ≤ u j +1 , the preceding inequality yields Z R n u qj +1 dσ ≤ C (cid:18)Z R n u qj +1 dσ (cid:19) qp − < ∞ . Thus, (cid:18)Z R n u qj +1 dσ (cid:19) p − − qp − ≤ C < ∞ . Using the Monotone Covergence Theorem and passing to the limit as j → ∞ in (3.7), we see that there exists u = lim j →∞ u j , such that u ∈ L q ( R n , dσ ),and the integral equation (2.5) holds. (cid:3) Lemma 3.6.
Let u ∈ L q ( R n , dσ ) be a nonnegative supersolution to theintegral equation (2.5) . Then (3.9) u q dσ ∈ L − ,p ′ ( R n ) . Proof.
Let dν = u q dσ . We need to show that, for all ϕ ∈ C ∞ ( R n ),(3.10) (cid:12)(cid:12)(cid:12)(cid:12)Z R n ϕ dν (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:18)Z R n |∇ ϕ | p dx (cid:19) p . It is easy to see that the above inequality is equivalent to(3.11) (cid:12)(cid:12)(cid:12)(cid:12)Z R n I g dν (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:18)Z R n | g | p dx (cid:19) p , for all g ∈ L p ( R n ), where I g is the Riesz potential of g of order 1. Byduality, (3.11) is equivalent to(3.12) Z R n ( I ν ) p ′ dx < ∞ . Using Wolff’s inequality (2.10), we deduce that (3.12) holds if and only if(3.13) Z R n W ,p ν dν < ∞ . Notice that since u ≥ W ,p ( u q dσ ) and u ∈ L q ( R n , dσ ) then Z R n W ,p ν dν = Z R n W ,p ( u q dσ ) u q dσ ≤ Z R n u q dσ < ∞ . Thus, (3.12) holds. This completes the proof of the lemma. (cid:3)
We will need a weak comparison principle which goes back to P. Tolks-dorf’s work on quasilinear equations (see, e.g., [PV08], Lemma 6.9, in thecase of renormalized solutions in bounded domains).
Lemma 3.7.
Suppose µ, ω ∈ M + ( R n ) ∩ L − ,p ′ ( R n ) . Suppose u and v are(quasicontinuous) solutions in L ,p ( R n ) of the equations − ∆ p u = µ and − ∆ p v = ω , respectively. If µ ≤ ω , then u ≤ v q.e.Proof. For every ϕ ∈ L ,p ( R n ), we have by Theorem 2.6,(3.14) Z R n |∇ u | p − ∇ u · ∇ ϕ dx = h µ, ϕ i = Z R n ϕ dµ, (3.15) Z R n |∇ v | p − ∇ v · ∇ ϕ dx = h ω, ϕ i = Z R n ϕ dω. Hence,(3.16) Z R n ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) · ∇ ϕ dx = Z R n ϕ dµ − Z R n ϕ dω. Since µ ≤ ω , it follows that, for every ϕ ∈ L ,p ( R n ), ϕ ≥
0, we have(3.17) Z R n ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) · ∇ ϕ dx ≤ . Testing (3.17) with ϕ = ( u − v ) + = max { u − v, } ∈ L ,p ( R n ), we obtain, I = Z R n ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) · ∇ ( u − v ) + dx ≤ . Let A = { x ∈ R n : u ( x ) > v ( x ) } , then I = Z A ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) · ∇ ( u − v ) dx ≤ . Note that ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) · ∇ ( u − v ) ≥ . Thus,0 ≤ Z A ( |∇ u | p − ∇ u − |∇ v | p − ∇ v ) · ∇ ( u − v ) dx = Z A ϕ ( dµ − dω ) ≤ . It follows that ∇ ( u − v ) = 0 a.e. on A . By Lemma 2.22 in [MZ97], for every a >
0, cap p { u − v > a } ≤ a p Z A |∇ ( u − v ) | p dx = 0 . Consequently, cap p ( A ) = 0, i.e., u ≤ v q.e. (cid:3) INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 13
We are now in a position to prove the main theorem of this section.
Theorem 3.8.
Let < p < n and < q < p − . Let σ ∈ M + ( R n ) , σ = 0 . Suppose that (1.8) holds. Then there exists a nontrivial solution w ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) to (1.1) . Moreover, w is a minimal solution, i.e., w ≤ u dσ -a.e. (q.e. for quasicontinuous representatives) for any nontrivialsolution u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) to (1.1) .Proof. We first show that there exists a solution w ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ )to (1.1). Applying Theorem 3.5, we conclude that there exists a solution v ∈ L q ( R n , dσ ) to the integral equation (2.5). By using a constant multiple c v in place of v, we can assume that v = K W ,p ( v q dσ ), where K is theconstant in Theorem 2.5. Then by Lemma 3.6 and Theorem 2.3, v q dσ ∈ L − ,p ′ ( R n ) , and v ≥ C K p − p − − q ( W ,p σ ) p − p − − q , where C is the constant in (2.6).We set w = c ( W ,p σ ) p − p − − q , dω = w q dσ, where c > w ≤ c CK p − p − − q v, it follows that, for c ≤ CK p − p − − q , we have w ≤ v . Hence, w ∈ L q ( R n , dσ ) , and ω ∈ L − ,p ′ ( R n ) . Then there exists a unique nonnegative solution w ∈ L ,p ( R n ) to the equa-tion − ∆ p w = ω , and || w || p − ,p = || ω || − ,p ′ . (See (2.8).) Moreover, by Theorem 2.5,0 ≤ w ≤ K W ,p ω ≤ K W ,p ( v q dσ ) = v. Consequently, by Lemma 3.6, w ∈ L q ( R n , dσ ) , and w q dσ ∈ L − ,p ′ ( R n ) . We deduce, using (3.3), w ≥ K W ,p ω = c qp − K W ,p (cid:16) ( W ,p σ ) q ( p − p − − q dσ (cid:17) ≥ c qp − κK ( W ,p σ ) p − p − − q = c qp − − κK w . Hence, for c ≤ ( K − κ ) p − p − − q , we have v ≥ w ≥ w .To prove the minimality of w , we will need c ≤ C , so we pick c so that(3.18) 0 < c ≤ min (cid:8) C K p − p − − q , ( K − κ ) p − p − − q , C (cid:9) . Let us now construct by induction a sequence { w j } j ≥ so that(3.19) ( − ∆ p w j = σ w qj − in R n , w j ∈ L ,p ( R n ) ∩ L q ( R n , dσ ) , ≤ w j − ≤ w j ≤ v, q.e., w qj − dσ ∈ L − ,p ′ ( R n ) , where sup j || w j || ,p < ∞ . We set dω j = w qj dσ , so that − ∆ p w j = ω j − , j = 1 , , . . . . Suppose that w , w , . . . , w j − have been constructed. As in the case j =1, we see that, since ω j − ∈ L − ,p ′ ( R n ), there exists a unique w j ∈ L ,p ( R n )such that − ∆ p w j = ω j − , and by (2.8), || w j || p ,p = || ω j − || p ′ − ,p ′ = Z R n w j w qj − dσ. By Theorem 2.5, we get w j ≤ K W ,p ω j − = K W ,p ( w qj − dσ ) . Using the inequality w j − ≤ v , we see that w j ≤ K W ,p ( v q dσ ) = v. Combining these estimates, we obtain || w j || p ,p = Z R n w j w qj − dσ ≤ Z R n v q dσ < ∞ . Consequently, { w j } is a bounded sequence in L ,p ( R n ). Notice that w j − ≤ w j by the weak comparison principle (Lemma 3.7), since ω j − ≤ ω j − , for j ≥ w = lim j →∞ w j ,and applying the weak continuity of the p -Laplace operator (Theorem 2.4),the Monotone Convergence Theorem, and Lemma 1.33 in [HKM06], we de-duce the existence of a nontrivial solution w ∈ L ,p ( R n ) to (1.1).We now prove the minimality of w . Suppose u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ )is any nontrivial solution to (1.1). Letting dµ = u q dσ , we have u ∈ L q ( R n , dσ ), and µ ∈ L − ,p ′ ( R n ) by Lemma 3.1. To show that u ≥ w ,notice that by Theorem 2.3, u ≥ C ( W ,p σ ) p − p − − q , where C is the constant in (2.6). By the choice of c in (3.18), we have w ≤ u , so that ω ≤ µ. Therefore, by the weak comparison principle w ≤ u q.e. Arguing by induction as above, we see that w j − ≤ w j ≤ u q.e. for j ≥
1. It follows that lim j →∞ w j = w ≤ u q.e., which proves that w is aminimal solution. (cid:3) By combining Lemma 3.1 and Theorem 3.8 we conclude the proof of theexistence part of the Theorem stated in the Introduction. In Sec. 5 below wewill establish the uniqueness part using the existence of a minimal solutionconstructed in Theorem 3.8.
INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 15 A -Laplace operators Let us assume that A : R n × R n → R n satisfies the following structuralassumptions: x → A ( x, ξ ) is measurable for all ξ ∈ R n ,ξ → A ( x, ξ ) is continuous for a.e. x ∈ R n , and there are constants 0 < α ≤ β < ∞ , such that for a.e. x in R n , and forall ξ in R n , A ( x, ξ ) · ξ ≥ α | ξ | p , |A ( x, ξ ) | ≤ β | ξ | p − , ( A ( x, ξ ) − A ( x, ξ )) · ( ξ − ξ ) > ξ = ξ , A ( x, λξ ) = λ | λ | p − A ( x, ξ ) , if λ ∈ R \{ } . Consider the equation(4.1) − div A ( x, ∇ u ) = µ in Ω , where µ ∈ M + (Ω), and Ω ⊆ R n is an open set. Let us use the de-composition µ = µ + µ s , where µ is absolutely continuous with respectto the p -capacity and µ s is singular with respect to the p -capacity. Let T k ( s ) = max {− k, min { k, s }} . We say that u is a local renomalized solution to (4.1) if, for all k > T k ( u ) ∈ W ,p loc (Ω), u ∈ L ( p − s loc for 1 ≤ s < nn − p ,Du ∈ L ( p − r loc (Ω) for 1 ≤ r < nn − , and Z Ω hA ( x, Du ) , Du i h ′ ( u ) φ dx + Z Ω hA ( x, Du ) , ∇ φ i h ( u ) φ dx = Z Ω h ( u ) φ dµ + h (+ ∞ ) Z Ω φ dµ s , for all φ ∈ C ∞ (Ω) and h ∈ W , ∞ ( R ) such that h ′ is compactly supported;here h (+ ∞ ) = lim t → + ∞ h ( t ) . In [KKT09], it is shown that every A -superhamonic function is locallya renormalized solution, and conversely, every local renormalized solutionhas an A -superharmonic representative. Consequently, we can work eitherwith local renormalized solutions, or equivalently with potential theoreticsolutions, or finite energy solutions in the case u ∈ L ,p (Ω). We note that,for finite energy solutions, Du coincides with the distributional gradient ∇ u ,and dµ = u q dσ is absolutely continuous with respect to the p -capacity aswas mentioned above.It is known that basic facts of potential theory stated in Sec. 2, includ-ing Wolff’s potential estimates [KM94], and the weak continuity principle[TW02], remain true for the A -Laplacian. From the above results it fol-lows that our methods work, with obvious modifications, not only for the p -Laplace operator, but for the general A -Laplace operator div A ( x, ∇ u ) aswell. In particular, the following more general theorem holds. Theorem 4.1.
Under the above assumptions on A ( x, ξ ) , together with theconditions of the Theorem stated in Sec. 1, the equation − div A ( x, ∇ u ) = σ u q has a solution u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) if and only if condition (1.8) holds. Uniqueness
In this section, we prove the uniqueness of finite energy solutions to (1.1).We employ a convexity argument using some ideas of Kawohl [Kaw00] (seealso [BeK02], [BF12]), together with the existence of a minimal solutionestablished above.
Theorem 5.1.
Let < p < ∞ and let < q < p − . Let σ ∈ M + ( R n ) .Suppose that there exists a nontrivial solution u ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) to (1.1) . Then such a solution is unique.Proof. Suppose u, v are nontrivial solutions to (1.1) which lie in L ,p ( R n ) ∩ L q loc ( R n , σ ). We first show that u = v dσ -a.e. implies that u = v as elementsof L ,p ( R n ).Indeed, suppose that u = v dσ -a.e., and set dµ = u q dσ = v q dσ , where µ ∈ M + ( R n ), and(5.1) − ∆ p u = − ∆ p v = µ. As usual, we assume that u, v are quasicontinuous representatives (see, e.g.,[HKM06], [MZ97]). Then by Lemma 3.1, u, v ∈ L q ( R n , dσ ), and Z R n W ,p µ dµ < + ∞ . By Wolff’s inequality (2.10), this means that µ ∈ L − ,p ′ ( R n ). It is wellknown ([MZ97], Sec. 2.1.5) that, for such µ , a finite energy solution to theequation − ∆ p u = µ is unique. (See also Lemma 3.7 above.) Hence, from(5.1) we deduce u = v q.e. and as elements of L ,p ( R n ).We next show that if u ≥ v dσ -a.e. then u = v dσ -a.e. By Theorem 2.3,it follows that u ( x ) > v ( x ) >
0, for all x ∈ R n . Testing the equations(5.2) Z R n |∇ u | p − ∇ u · ∇ φ dx = Z R n u q ϕ dσ, φ ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) , (5.3) Z R n |∇ v | p − ∇ v · ∇ ψ dx = Z R n v q ψ dσ, ψ ∈ L ,p ( R n ) ∩ L q loc ( R n , dσ ) , with φ = u, ψ = v , respectively, we obtain Z R n |∇ u | p dx = Z R n u q dσ, Z R n |∇ v | p dx = Z R n v q dσ. Let λ t ( x ) = (cid:16) (1 − t ) v p ( x ) + tu p ( x ) (cid:17) p . INITE ENERGY SOLUTIONS OF QUASILINEAR EQUATIONS 17
Using convexity of the Dirichlet integral R R n |∇ u | p dx in u p [Kaw00] (see alsothe proof of Lemma 2.1 in [BF12]), we estimate, for all t ∈ [0 , Z R n |∇ λ t ( x ) | p dx ≤ (1 − t ) Z R n |∇ v | p dx + t Z R n |∇ u | p dx = t (cid:18)Z R n |∇ u | p dx − Z R n |∇ v | p dx (cid:19) + Z R n |∇ v | p dx. Thus, Z R n |∇ λ t ( x ) | p − |∇ λ ( x ) | p t dx ≤ Z R n u q dσ − Z R n v q dσ. Using the inequality | a | p − | b | p ≥ p | b | p − b · ( a − b ) , a, b ∈ R n , we deduce |∇ λ t | p − |∇ λ | p ≥ p |∇ λ | p − ∇ λ · ( ∇ λ t − ∇ λ ) . Notice that λ = v , and consequently, for all t ∈ (0 , p Z R n |∇ v | p − ∇ v · ∇ ( λ t − λ ) t dx ≤ Z R n u q d σ − Z R n v q d σ. Testing (5.3) with ψ = λ t − λ ∈ L ,p ( R n ), we obtain Z R n |∇ v | p − ∇ v · ∇ ( λ t − λ ) dx = Z R n v q ( λ t − λ ) dσ. Hence, by (5.4), for all t ∈ (0 , p Z R n v q λ t − λ t dσ ≤ Z R n u q d σ − Z R n v q d σ. Clearly, λ t ≥ λ , since u ≥ v . Applying Fatou’s lemma, we obtain Z R n v q u p − v p v p − dσ ≤ lim inf t → p Z R n v q λ t − λ t dσ. Combining this and (5.5) yields Z R n ( v q u p v p − − v q ) dσ ≤ Z R n u q d σ − Z R n v q d σ. Therefore, canceling the second terms on both sides, and taking into accountthat u ≥ v dσ -a.e., we arrive at0 ≥ Z R n ( v q u p v p − − u q ) dσ = Z R n v q u p − u q v p − v p − dσ = Z R n v q u q ( u p − − q − v p − − q ) v p − dσ ≥ . Hence, u = v dσ -a.e.We now complete the proof of the uniqueness property. Suppose that u and v are nontrivial finite energy solutions to (1.1). Then min ( u, v ) ≥ wdσ -a.e., where w is the nontrivial minimal solution constructed in Theorem w = u = v dσ -a.e., and also as elementsof L ,p ( R n ). (cid:3) References [ABV10] H. Abdel Hamid and M.-F. Bidaut-V´eron,
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Department of Mathematics, University of Missouri, Columbia, Missouri65211, USA
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