QUASILINEAR ELLIPTIC EQUATIONS WITH SUB-NATURAL GROWTH AND NONLINEAR POTENTIALS

Transcription

1 QUASILINEAR ELLIPTIC EQUATIONS WITH SUB-NATURAL GROWTH AND NONLINEAR POTENTIALS A Dissertation presented to the Faculty of the Graduate School at the University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by CAO TIEN DAT Under the supervision of Dr. IGOR E. VERBITSKY JULY 205

2 c Copyright by CAO TIEN DAT 205 All Rights Reserved

3 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled: QUASILINEAR ELLIPTIC EQUATIONS WITH SUB-NATURAL GROWTH AND NONLINEAR POTENTIALS presented by CAO TIEN DAT, a candidate for the degree of Doctor of Philosophy and hereby certify that, in their opinion, it is worthy of acceptance. Professor Igor E. Verbitsky Professor Steven Hofmann Professor Loukas Grafakos Professor Stephen Montgomery-Smith Professor David Retzloff

4 ACKNOWLEDGMENTS First and foremost, I wish to express my deepest gratitude to my adviser, Professor Igor E. Verbitsky, for introducing to me this topic, sharing his ideas and for his continuous advice, support and guidance during my Ph.D. program. I am very grateful to Professors Steven Hofmann, Loukas Grafakos and Stephen Montgomery-Smith for their interest and support. I would also like to thank Professors Nguyen Cong Phuc and Benjamin J. Jaye for some useful discussions. I am also grateful to the Mathematics department at the University of Missouri for the financial support. Finally, I wish to thank my family and my wife for their constant encouragement and support. ii

5 TABLE OF CONTENTS ACKNOWLEDGMENTS ABSTRACT ii v CHAPTER Introduction Preliminaries Notations, definitions Wolff potential estimates Finite energy solutions Main results Existence and minimality of finite energy solutions Uniqueness Weak solutions and intrinsic potentials of Wolff type Main results Solutions of the integral equations Weighted norm inequalities and intrinsic potentials K α,p,q σ Solution in L q (R n, dσ) Solution in L q loc (Rn, dσ) Solution in L +q loc (Rn, dσ) Proofs of Theorem 4., 4.2 and Theorem iii

6 4.4 Example Equations with singular gradient terms Extension of the Brezis-Kamin theorem Main results Capacity condition and Wolff potential estimates Solutions of the nonlinear integral equations Inhomogeneous problems Homogeneous problems Proofs of Theorem 5., 5.2, 5.3 and Theorem Radial case APPENDIX A Uniqueness of solutions to quasilinear PDEs B Weak compactness in L,p 0 (R n ) BIBLIOGRAPHY VITA iv

7 ABSTRACT In this thesis, we study quasilinear elliptic equations of the type (0.) p u = σu q in R n, where p u = ( u u p 2 ) is the p-laplacian, < p <, and σ 0 is an arbitrary locally integrable function, or measure, in the sub-natural growth case 0 < q < p. Necessary and sufficient conditions on σ for the existence of finite energy and weak solutions to (0.) are given. Sharp global pointwise estimates of solutions are obtained as well. We also discuss the uniqueness and regularity properties of solutions. As a consequence, characterization of solvability of the equation (0.2) p v = b v p v + σ in R n, where b > 0, is deduced. Our main tools are Wolff potential estimates, dyadic models, and related integral inequalities. Special nonlinear potentials of Wolff type associated with sublinear problems are constructed to obtain sharp bounds of solutions. We also treat equations with the fractional Laplacians ( ) α. Our approach is applicable to more general quasilinear A-Laplace operators diva(x, ) as well as the fully nonlinear k-hessian operators. v

8 Chapter Introduction This work is concerned with quasilinear problems of the following type: (.) p u = σu q in R n, lim inf x u(x) = 0, u > 0, where p u = ( u u p 2 ) is the p-laplacian, < p <, and σ 0 is an arbitrary locally integrable function, or measure, in the case 0 < q < p (sub-natural growth rate). We will give necessary and sufficient conditions for the existence of finite energy and weak solutions to (.). Sharp global pointwise bounds along with regularity properties of solutions are obtained as well. We identify crucial integral inequalities and introduce new nonlinear potentials of Wolff type adapted for these problems. We

9 also study the fractional Laplacian equation (.2) where 0 < α < n 2 and 0 < q <. ( ) α u = σu q in R n, lim inf x u(x) = 0, u > 0, In the classical case p = 2 and 0 < q <, equation (.), or (.2) with α =, serves as a model sublinear elliptic problem. Such problems were studied by H. Brezis and S. Kamin in [BK92], where they gave necessary and sufficient conditions for the existence of bounded solutions. In particular, they proved that equation (.) has a bounded solution u if and only if I 2 σ L (R n ), where I 2 σ is the Riesz potential of order 2 (Newtonian potential) defined by (.3) I 2 σ(x) = 0 σ(b(x, r)) dr r n 2 r = c n R n x y n 2, x Rn. Moreover, such a solution u is unique and there is a constant c = c(n, q) > 0 so that c (I 2 σ(x) q u(x) c I2 σ(x), x R n. Both the lower and upper estimates are sharp in a sense, as was pointed out in [BK92]. However, there is a substantial gap between them. The difficulty here is to find matching lower and upper bounds of solutions and getting such a result is our motivation. Analogous sublinear problems in bounded domains Ω R n for various classes of σ have been extensively studied. In particular, Boccardo and Orsina [BO96], [BO2], and Abdel Hamid and Bidaut-Véron [AHBV0] gave sufficient conditions for the 2

10 existence of solutions under the assumption σ L r (Ω). Earlier results, under more restrictive assumptions on σ are due to Krasnoselskii [Kra64], and Brezis and Oswald [BrOs86] (see also the literature cited there). We will consider two different types of solutions : finite energy and weak solutions. The first one is a solution which lies in L q loc (Rn, dσ) and belongs to the homogeneous Sobolev (or Dirichlet) space L,p 0 (R n ) defined in Chapter 2. The second one only requires solutions in L q loc (Rn, dσ). Finding weak solutions is considered to be much more complicated than that of finite energy solutions. In both cases, we are able to obtain necessary and sufficient conditions for the existence of such solutions. We employ powerful Wolff potential estimates due to Kilpeläinen and Malý, Trudinger and Wang, and Labutin in [KM94, La02, TW02b, KuMi4]. This makes it possible to replace the p-laplacian p in the model problem (.) by a more general quasilinear operator diva(x, ) with bounded measurable coefficients, under standard 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, La02] (see also [PV09], [JV2]), and the fractional Laplacian ( ) α as well. The Wolff potential W α,p σ of a nonnegative locally Borel measure σ on R n, is defined, for < p < and 0 < α < n, by ([HW83, AH96]): p (.4) W α,p (x) = 0 ( σ(b(x, t)) t n αp dt t. Here B(x, t) = {y R n : x y < t} is a ball centered at x R n of radius t > 0. To study (.), we make use of the pointwise estimates of solutions to quasilinear equations in terms of Wolff potentials, which were discovered by Kilpeläinen and Malý 3

11 in [KM94, Ki02]. One of their main theorems states that if U 0 is a p-superharmonic (or locally renormalized, see Chapter 2) solution to the equation (.5) p U = σ in R n, lim inf x U(x) = 0, then there exists a constant K > 0 which depends only on p and n such that (.6) K W,pσ(x) U(x) K W,p σ(x), x R n. Moreover, a solution U to (.5) exists if and only if < p < n and W,p σ + (see [PV08]), or equivalently, (.7) ( σ(b(0, t)) t n p dt t < +. We study the general framework of integral equations which are closely related to nonlinear elliptic PDE mentioned above, (.8) u = W α,p (u q dσ) in R n, u 0, where < p <, 0 < α < n. When α =, it gives the corresponding p-laplacian, p and α = 2k, p = k + correspond to the k-hessian operator. We notice that in k+ the special case p = 2, (.8) brings the fractional Laplace equation (.2) into the equivalent integral form u = c(α,n) I 2α(u q dσ), where I 2α µ is the Riesz potential of order 2α with dµ = u q dσ, defined by (2.3). For the sake of simplicity, the constant c(α, n) will be dropped. 4

12 We are also interested in the supersolution of (.8), which is a solution of the following nonlinear integral inequality (.9) u W α,p (u q dσ) in R n, u 0. Our first result states that if u is a nontrivial supersolution to either equation (.) or integral equation (.8) with α =, then u satisfies the following lower estimate (.0) u(x) C (W,p σ(x)) q, x R n, where C > 0 is a constant depending only on n, p and q. Consequently, a necessary condition for the existence of a nontrivial solution to (.) is that W,p σ +, i.e., (.7) holds. Estimate (.0) is one of our key tools to obtain the main results in this work. Concerning the finite energy solutions, we make an observation due to Brezis and Browder, which says that, if there is a solution u L,p 0 (R n ) to (.), then u belongs to L +q (R n, dσ) globally. Combining this fact with (.0), we arrive at a necessary condition (.) R n (W,p σ) (+q)() q dσ <. As shown in Chapter 3, condition (.) holds if and only if there exists a constant C > 0 such that the following weighted norm inequality holds, (.2) ( R n ϕ +q dσ +q C ϕ L p (R n ), ϕ C 0 (R n ). 5

13 Some equivalent characterizations for (.) will be discussed in Chapter 3 as well. One might ask whether (.) is sufficient for the existence of finite energy solutions to (.). An affirmative answer to this question is given in the same chapter. We remark that (.) yields W,p σ +, which is the necessary condition mentioned above. Moreover, we are able to show that such a finite energy solution is unique. When studying weak solutions, it turns out that (.) is closely related to the following important integral inequality (.3) ( R n ϕ q dσ q κ p ϕ L (R n ), for all test functions ϕ such that p ϕ 0, lim inf x ϕ(x) = 0. Here κ denotes the best constant in (.3). Such an inequality (.3) corresponds to the end-point case of the (L p, L q )-trace inequality for p >, and is less studied when comparing with the well-studied inequality (.2). Having (.6) in hand, we see that (.3) is equivalent to the inequality (.4) W,p ν L q (R n,dσ) κ(ν(r n ), ν M + (R n ), here κ denotes the least constant in the above inequality. We will need a local version of the preceding inequality, where the measure σ = σ B is restricted to a ball B in R n : (.5) W,p ν L q (dσ B ) κ(b)(ν(r n ), ν M + (R n ), where κ(b) is the best constant in this inequality. These constants are used as 6

14 building blocks in our key tool, a new nonlinear potential of Wolff type, (.6) K,p,q σ(x) = 0 (κ(b(x, s)) q() q s n p ds s, x Rn. This intrinsic nonlinear potential of Wolff type has never appeared before and will control the solutions to (.). Similarly to (.0), we also have an estimate for a nontrivial supersolution u to (.) as follows (.7) u(x) C K,p,q σ(x), x R n, where C > 0 is a constant depending only on n, p, and q. Consequently, another necessary condition is that K,p,q σ, or equivalently, (.8) (κ(b(0, s)) q() q s n p ds s <. The potential K,p,q σ together with the usual Wolff potential W,p σ provides sharp global pointwise estimates of the solutions to (.). Using both of them allow us to bridge the gap in the estimates of Brezis-Kamin. In particular, as we will show in Chapter 4, (.7) and (.8) are necessary and sufficient for the existence of a miminal weak solution u to (.). Moreover, such a solution u has matching lower and upper estimates as follows ( ) ) (.9) c K,p,q σ + (W,p σ) q u c (K,p,q σ + (W,p σ) q, where c > 0 is a constant depending only on n, p, and q. 7

15 If one wishes to have a solution with W,p loc -regularity, then together with (.7) and (.8), a local version of (.) is needed, i.e., (.20) B (W,p σ B ) (+q)() q dσ <, for all balls B in R n. We will also see that (.20) is necessary in order to have a solution in W,p loc (Rn ) to (.). For the fractional Laplace equation (.2), let K(B) denote the least constant in the integral inequality (.2) I 2α ν L q (dσ B ) K(B)ν(R n ), ν M + (R n ). We define the corresponding nonlinear potential of Wolff type by (.22) K α,q σ(x) = 0 K(B(x, s)) q q s n 2α ds s, x Rn. To ensure that both K α,q σ and I 2α σ are not indentically infinite, the following condition should hold (.23) K(B(0, s)) q q s n 2α ds s + σ(b(0, s)) ds s n 2α s <. As a consequence of our main results, (.23) is necessary and sufficient for the existence of a solution u to the sublinear fractional Laplace equation (.2), and there exists a minimal solution u which satisfies ( ) ) (.24) c K α,q σ + (I 2α σ q u c (K α,q σ + (I 2α σ q. 8

16 We also observe that if there is a nontrivial p-superharmonic supersolution to (.) or (.8) with α =, then the measure σ is absolutely continuous with respect to p-capacity cap p ( ), i.e., σ(e) = 0 if cap p (E) = 0, for all compact sets E in R n, where (.25) cap p (E) = inf{ f p L p : f on E, f C 0 (R n )}. In Chapter 5, we study problem (.) under the assumption that (.26) σ(e) C(σ) cap p (E) for all compact sets E R n, where C(σ) is a positive constant. Such a condition was considered in [JV0, JV2, Maz]. Making use of the sub-supersolutions method, we obtain the main result in this direction which states that if both (.7) and (.26) hold, then there exists a distributional solution u W,p loc (Rn ) to (.) and u satisfies ) (.27) C (W q,p σ u C ) (W,p σ + (W,p σ) q, where C > 0 depends only on n, p, q, and C(σ). Both estimates are sharp as in the Brezis-Kamin theorem. In the case p = 2, 0 < q <, we remark that if I 2 σ L (R n ) (considered by Brezis-Kamin), then by a well known result it follows that (.26) holds with p = 2. Hence, by (.27), there exists a bounded solution u to (.) with p = 2 and u satisfies c (I 2 σ q u c I2 σ, where c = c(n, q). From this observation we see that our condition is weaker than 9

17 that of Brezis-Kamin. Moreover, our results extend to possibly unbounded solutions as well. Also, under conditions (.7) and (.26), we obtain the existence and pointwise estimates for a solution u to (.) with positive lower bound, i.e., equation of the type (.28) p u = σu q in R n, lim inf x u(x) = r, where r > 0, and u has matching upper and lower bounds as follows ) (.29) c (r q + W,p σ(x) ( ) q u(x) c r + W,p σ(x), x R n. From (.27), we can deduce, using (.6) and (.0), ( ) ) (.30) W,p (W,p σ) ()q q dσ κ (W,p σ + (W,p σ) q < a.e., where κ = κ(n, p, q) is a positive constant. It turns out that the preceding condition is enough to have a solution which satisfies (.27). So (.30) is not only necessary but also sufficient for the existence of solutions to (.) satisfying (.27). In the case p = 2 and σ is radial, we establish bilateral bounds of solution u to 0

18 (.) as follows. (.3) c ( u(x) c x n 2 ( ( x n 2 y < x y (n 2)q ( y < x q + ( y x ( q + y (n 2)q y x ) ) q y n 2 q y n 2 ), x R n. Let us make a remark that we will be referring to (.) with < p < and 0 < q < p, as well as other nonlinear equations where analogous phenomena occur in a natural way, as sublinear problems in general. One of the main features that distinguishes them from the case p p is the absence of any smallness asumptions on σ. Simultaneously with (.), we consider in Chapter 4 the equation with the singular natural growth in the gradient term, which is closely related to (.) : (.32) p v = b v p v + σ in R n, lim inf x v = 0. where b is a positive constant defined by (.33) b = q(p ) p q, 0 < q < p. Equation (.32) with p = 2 in a bounded domain Ω R n has been studied by D. Arcoya et al. in [ABLP0] and B. Abdellaoui et al. in [AGPW], in which they gave sufficient conditions for the existence of solutions in certain Sobolev spaces under the assumption σ L s (Ω) for some s > (see also the literature given there).

19 It is well known that formally the substitution (.34) v = p p q u q reduces (.) to (.32), and vice versa. However, in general, this substitution fails for some solutions u and v since some certain singular measures can arise, as was first noticed by Ferone and Murat [FM00] who studied a similar phenomenon in the case q = p. Nevertheless, a careful justification enables us to give necessary and sufficient conditions for the existence of weak and finite energy solutions to (.32), and obtain pointwise estimates of such solutions as well. As we will demonstrate in Chapter 4, if u is a solution to (.) then v is a solution to (.32) via substitution (.34); but in the opposite direction, if v is a weak solution to (.32) then u is only a supersolution to (.), which is enough for our purposes. The content of this thesis is as follows. In Chapter 2 we introduce basic definitions and notations, along with several useful results on quasilinear equations and Wolff potentials estimates that will be used throughout the text. In Chapter 3 we study finite energy solutions to (.) and prove the uniqueness property of such solutions. Chapter 4 is devoted to a study of weak solutions to equations (.) and (.32). The intrinsic nonlinear potentials of Wolff type are introduced in this chapter as well. Chapter 5 is concerned with equation (.) under the assumption (.26). We provide, in Appendix A, the proof of uniqueness of solutions to (.5) in a bounded domain when σ is absolutely continuous with respect to cap p ( ). Finally, the weak compactness property of L,p 0 (R n ) is given in Appendix B. The content of this dissertation is taken from the joint works with Professor Igor E. Verbitsky [CV4a, CV4b, CV5]. 2

20 Chapter 2 Preliminaries 2. Notations, definitions Given an open set Ω R n, we denote by M + (Ω) the class of all nonnegative Borel measures in Ω which are finite on compact subsets of Ω. If σ M + (Ω), the σ-measure of a measurable set E Ω is denoted by E σ = σ(e) = dσ. We write A B if E there are two universal constants c and c 2 such that c A B c 2 A. For p > 0 and σ M + (Ω), we denote by L p (Ω, dσ) (L p loc (Ω, dσ), respectively) the space of all measurable functions f such that f p is integrable (locally integrable) with respect to σ. For f L p (Ω, dσ), we set ( f L p (Ω,dσ) = Ω f p dσ p. When dσ = dx, we write L p (Ω) (respectively L p loc (Ω)), and denote Lebesgue measure of E R n by E. 3

21 The Sobolev space W,p (Ω) (W,p loc (Ω), respectively) is the space of all functions f such that f L p (Ω) and f L p (Ω) (f L p loc (Ω) and f Lp loc (Ω), respectively). As usual, W,p 0 (Ω) is the closure of C 0 (Ω) with respect to the Sobolev norm f,p = f L p + f L p. By L,p 0 (Ω) we denote the homogeneous Sobolev space, i.e., the space of functions f W,p loc (Ω) such that f Lp (Ω), and f ϕ j L p (Ω) 0 as j for a sequence ϕ j C 0 (Ω). When < p < n and Ω = R n, we will identify L,p 0 (R n ) with the space of all functions f W,p loc (Rn ) such that f L np n p (R n ) and f L p (R n ). L,p 0 (R n ), the norm f,p = f L p is equivalent to For f f L np n p (R n ) + f L p (R n ). It is easy to see that C 0 (R n ) is dense in L,p 0 (R n ) with respect to this norm (see, e.g., [MZ97], Sec..3.4). The dual Sobolev space L,p (Ω) = L,p 0 (Ω) is the space of distributions ν D (Ω) such that ν,p = sup f, ν f,p < +, where the supremum is taken over all f L,p 0 (Ω), f 0. We write ν L,p loc (Ω) if ϕ ν L,p (Ω), for every ϕ C0 (Ω). We denote by W,p (Ω) the dual space of W,p 0 (Ω). We say ν W,p (Ω) if ϕ ν W,p (Ω), for every ϕ C 0 (Ω). We will need Wolff s inequality [HW83] (see also [AH96], Sec. 4.5) in the case loc 4

22 Ω = R n for ν M + (R n ): (2.) c ν p,p W,p ν dν c ν p,p, R n where < p < n, and c is a positive constant which depends only on n and p. There is a local version of Wolff s inequality (see [AH96], Theorem 4.5.5): (2.2) ν M + (R n ) W,p loc (R n ) W,p ν B dν B <, for all balls B, B where B = B(x, R), and ν B = ν B. The following theorem is due to Brezis and Browder [BB79] (see also [MZ97], Theorem 2.39). Theorem 2.. Let < p < n. Suppose u L,p 0 (R n ), and µ M + (R n ) L,p (R n ). Then u L (R n, µ) (for a quasicontinuous representative of u), and (2.3) µ, u = u dµ. R n We observe that if, under the assumptions of this theorem, p u = µ, then it follows (see [MZ97], Theorem 2.34) (2.4) µ, u = u dµ = u p,p = µ p,p. R n We next define the A-Laplace operator as the mapping A : R n R n R n satisfies the following structural assumptions: x A(x, ξ) is measurable for all ξ R n, 5

23 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 for all ξ in R n, A(x, ξ) ξ α ξ p, A(x, ξ) β ξ, (A(x, ξ ) A(x, ξ 2 )) (ξ ξ 2 ) > 0 if ξ ξ 2, A(x, λξ) = λ λ p 2 A(x, ξ), if λ R\{0}. The p-laplace operator corresponds to the choice A(x, ξ) = ξ p 2 ξ. For u W,p loc (Ω), we define the p-laplacian p ( < p < ) in the distributional sense, i.e., for every ϕ C 0 (Ω), (2.5) p u, ϕ = div( u p 2 u, ϕ = u p 2 u ϕ dx. Ω We will extend the usual distributional definition of solutions u of p u = µ, where µ W,p loc (Ω), to u not necessarily in W,p loc (Ω). We will understand solutions in the following potential-theoretic sense using p-superharmonic functions, which is equivalent to the notion of locally renormalized solutions in terms of test functions (see [KKT09]). A function u W,p loc (Ω) is called p-harmonic if it satisfies the homogeneous equation p u = 0. Every p-harmonic function has a continuous representative which coincides with u a.e. (see [HKM06]). As usual, p-superharmonic functions are defined via a comparison principle. We say that u : Ω (, ] is p-superharmonic if u is lower semicontinuous, is not identically infinite in any component of Ω, and, whenever D Ω and h C(D) is 6

24 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 truncations T k (u) = min(k, max(u, k)) do, for all k > 0. In addition, T k (u) are supersolutions, i.e., div( T k (u) p 2 T k (u)) 0, in the distributional sense. 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 representative which coincides with u quasieverywhere (q.e.), i.e., everywhere except for a set of p-capacity zero (see [HKM06]). We will assume that u is always chosen to be quasicontinuous. Let u be p-superharmonic, and let r < n. Then n Du, and consequently Du p 2 Du, belongs to L r loc (Ω) [KM92]. This allows us to define a nonnegative distribution p u for each p-superharmonic function u by (2.6) p u, ϕ = Ω Du p 2 Du ϕ dx, for all ϕ C 0 (Ω). Then by the Riesz representation theorem there exists a unique measure µ[u] M + (Ω) so that p u = µ[u], where µ[u] is called the Riesz measure of u. Definition 2.2. For ω M + (Ω), u is said to be a (p-superharmonic) solution to the equation (2.7) p u = ω in Ω 7

25 if u is p-superharmonic in Ω, and µ[u] = ω (see [KM92], [KM94], [Ki02]). Thus, if σ M + (Ω), then u 0 is a (weak) solution to the equation (2.8) p u = σu q in Ω if u is p-superharmonic in Ω, u L q loc (Ω, dσ), and dµ[u] = uq dσ. Alternatively, we will use the framework of locally renormalized solutions. This notion introduced by Bidaut-Véron [BiVe03], following the development of the theory of renormalized solutions in [MMOP99], is well suited for our purposes. As was shown recently in [KKT09], for ω M + (Ω) it coincides with the notion of a p-superharmonic solution in Definition 2.2. In particular, a p-superharmonic function u 0 satisfying (2.7) is a locally renormalized solution defined in terms of test functions (see [KKT09], Theorem 3.5). This means that, for all ϕ C0 (Ω) and h W, (R + ) with h having compact support, we have (2.9) Du p h (u) ϕ dx + Du p 2 Du ϕ h(u) dx = h(u)ϕ d ω. Ω Ω Ω The converse is also true, i.e., if u is a locally renormalized solution to (2.7) then there exists a p-superharmonic representative ũ = u a.e. We will call such solutions of (2.9) with dω = u q dσ (locally) renormalized, p- superharmonic, or simply solutions, of (2.8). Definition 2.3. A function u 0 is called a (renormalized) supersolution to (2.8) if 8

26 u is p-superharmonic in Ω, u L q loc (Ω, dσ), and (2.0) Du p 2 Du ϕ dx u q ϕ dσ, ϕ C0 (Ω), ϕ 0. Ω Ω As we will show below, supersolutions to (.) in the sense of Definition 2.3 are closely related to supersolutions associated with the integral equation (.8), i.e., u L q loc (Rn, dσ) such that (2.) u W α,p (u q dσ) dσ-a.e., in the case α =. We will employ some fundamental results of the potential theory of quasilinear elliptic equations. First, let us state a useful convergence result in [KM92], Theorem.7. Theorem 2.4. Suppose {u j } j is a sequence of nonnegative p-superharmonic functions in an open set Ω. Then there is a subsequence {u jk } k of u j which converges almost everywhere to a nonnegative function u which is either p-superharmonic or identically infinite in each component of Ω and Du jk Du a.e. in {u < }. The following important weak continuity result [TW02b] will be used to prove the existence of p-superharmonic solutions to quasilinear equations. Theorem 2.5. Suppose {u j } is a sequence of nonnegative p-superharmonic functions that converges a.e. to a p-superharmonic function u in an open set Ω. Then µ[u j ] 9

27 converges weakly to µ[u], i.e., for all ϕ C 0 (Ω), lim ϕ dµ[u j ] = ϕ dµ[u]. j Ω Ω The next result [KM94] is concerned with global pointwise estimates of nonnegative p-superharmonic functions in terms of Wolff s potentials discussed in the Introduction. Theorem 2.6 ([KM94]). Let < p <, and let u be a p-superharmonic function in R n with lim inf x u(x) = 0. (i) If p < n and ω = µ[u], then (2.2) K W,pω(x) u(x) K W,p ω(x), x R n, where K is a positive constant depending only on n and p. (ii) In the case p n, it follows that u 0. For 0 < α < n and σ M + (R n ), the Riesz potential of order α is defined by (2.3) I α σ(x) = 0 σ(b(x, r)) dr r n α r = c n R n x y n α, x Rn. For E R n, we define the Riesz capacity of E by (2.4) cap α,p (E) = inf{ f p L p (R n ) : f Lp (R n ), f 0, I α f on E }. 20

28 We have, for all compact sets E R n, (2.5) c cap,p(e) cap p (E) c cap,p (E), where c = c(n, p) (see [MZ97]). We next define the truncated Wolff potential W R α,pσ, where R > 0, by W R α,pσ(x) = R 0 ( σ(b(x, t)) t n αp dt t, x Rn. In some instances, it is more convenient to work with the dyadic version of Wolff potential (2.6) W α,p σ(x) = Q D [ ] σ(q) αp χq (x), Q n where D is the set of dyadic cubes in R n. The shifted version of the Wolff s potential, for µ M + (R n ), t R n, is defined by W d,t α,pµ(x) = [ µ(q) Q D t Q αp n ] χq (x), where Q now denotes a shifted dyadic cube in the lattice D t = D+t = {Q+t} Q D. Let us state a useful dyadic shifting lemma which goes back to the papers [FS7, GJ82]. Lemma 2.7. Let R > 0, there exist constants c, c > 0 depending only on n such that for all x R n : (2.7) W R α,pµ(x) c R n t cr W d,t α,pµ(x)dt. 2

29 See, for instance, [COV00]. We will need the following Wolff s inequality [HW83] (see also [AH96], Sec. 4.5) which gives precise estimates of the energy associated with the Wolff potential: Theorem 2.8. Suppose < p <, 0 < α < n p, and σ M + (R n ). Then there exists a constant C > 0 depending only on p, α, and n such that (2.8) C (I α σ) p dx W α,p σ dσ C R n R n (I α σ) p dx, R n where p + p =. 2.2 Wolff potential estimates We start with some useful estimates for Wolff potentials. Throughout this section we will assume that σ M + (R n ) and σ 0. Lemma 2.9. Suppose < p <, 0 < α < n p, and σ M + (R n ). Let s = min (, p ). Then there exists a positive constant c which depends only on n, p, and α such that, for all x R n and R > 0, (2.9) ( σ(b(x, r)) c R R r n αp dr r ( inf W α,pσ B(x,R) B(x, R) ( σ(b(x, r)) dr c r n αp r. B(x,R) [W α,p σ(y)] s s dy Proof. Without loss of generality we can assume that x = 0. We first prove the last 22

30 estimate in (2.9). Clearly, [W α,p σ(y)] s dy I + I 2, B(0, R) B(0,R) where I = I 2 = B(0, R) B(0, R) B(0,R) B(0,R) ( R ( σ(b(y, r)) 0 r n αp ( ( σ(b(y, r)) R r n αp ) s dr dy, r ) s dy. dr r To estimate I 2, notice that since B(y, r) B(0, 2r) for y B(0, R) and r > R, it follows ( ( σ(b(0, 2r)) I 2 R r n αp ) s dr. r To estimate I, suppose first that p 2 so that s =. Then using Fubini s theorem and Jensen s inequality we deduce R ( I c 0 B(0, R) B(0,R) Using Fubini s theorem again, we obtain B(0,R) σ(b(y, r)) dy B(0,2R) σ(b(y, r)) dy dr r. n αp + B(y, r) dσ = B(0, ) r n σ(b(0, 2R)). Hence, there is a constant c = c(n, p, α) such that R I cr n σ(b(0, 2R)) = cr αp n σ(b(0, 2R)) 0 r αp dr ( σ(b(0, 2r)) c R r n αp dr r. 23

31 Notice that this is the same estimate we have deduced above for I 2 with s =. Let us now estimate I for < p < 2 and s = p. In this case, we will use the following elementary inequality: for every R > 0, ( R ( φ(r) 0 r γ ) dr 2R c(p, γ) r 0 φ(r) dr r γ r, where γ > 0, < p < 2, and φ is a non-decreasing function on (0, ). Applying the preceding inequality with φ(r) = σ(b(y, 2r)) and γ = n αp, and estimating as in the case p 2, using Fubini s theorem again, we obtain: I c 2R σ(b(y, r)) dr B(0, R) B(0,R) 0 r n αp r dy c R n σ(b(0, 2R)) 2R ( ( σ(b(0, 2r)) c R r n αp 0 r α dr = c R n+αp σ(b(0, 2R)) ) dr, r where c denotes different constants depending only on n, p, α. Combining the estimates for I and I 2, we arrive at ( (W α,p σ) s dy c B(0, R) B(0,R) R ( σ(b(0, 2r)) r n αp ) s dr. r Making the substitution ρ = 2r in the integral on the right-hand side completes the proof of the upper estimate in (2.9). To prove the lower estimate, notice that W α,p σ(y) 2R ( σ(b(y, r)) r n αp dr r = c R ( σ(b(y, 2ρ)) ρ n αp dρ ρ. 24

32 Since B(y, 2ρ) B(0, ρ) for y B(0, R) and ρ > R, there exists c = c(n, p, α) > 0 such that inf W α,pσ c B(0,R) R ( σ(b(0, ρ)) ρ n αp dρ ρ. Corollary 2.0. Suppose < p <, 0 < α < n p, and σ M + (R n ). (i) W α,p σ + if and only if (2.20) ( σ(b(0, r)) r n αp dr r <. (ii) Condition (2.20) implies (2.2) t ( σ(b(x, r)) r n αp dr r <, x Rn, t > 0. (iii) If (2.20) holds, then W α,p σ L s loc (dx), where s = min (, p ), and (2.22) lim inf x W α,pσ(x) = 0. Proof. We first verify statement (ii). Suppose (2.20) holds. We may assume x 0, since for x = 0 (2.2) is obvious. Clearly, B(x, r) B(0, 2r) for x < r, and hence, I x := x ( σ(b(x, r)) r n αp dr r x ( σ(b(0, 2r)) r n αp dr r <. It follows that (2.2) holds for t x. If t < x, then t ( σ(b(x, r)) r n αp dr r = x t 25 ( σ(b(x, r)) r n αp dr r + I x <,

33 since in the first integral B(x, r) B(0, 2 x ). Thus, (2.2) holds for all x and t > 0. It remains to prove (2.22), since the other statements of Corollary 2.0 are immediate from (2.9) and (2.2). Suppose that (2.20) holds. For R > 0, let A R = { R 2 < x < R}. Then by the upper estimate of Lemma 2.9 (with x = 0), inf W α,pσ(x) inf x >R/2 A R c ( W α,p σ(x) (W α,p σ) s dx A R A R ( σ(b(0, r)) dr r n αp r, R where c does not depend on R. Since the right-hand side of the preceding inequality tends to zero as R, we see that (2.22) holds. It is easy to see that if ω M + (R n ), and u W,p loc (Rn ) is a weak solution to the equation p u = ω, then ω W,p loc (R n ). The converse statement is less obvious, and we were not able to find it in the literature. In the next lemma, for the sake of completeness, we give a proof in the case ω 0 using a series of Caccioppoli-type inequalities. s Lemma 2.. Suppose < p < n, and ω M + (R n ) W,p loc (R n ). If u 0 is a p- superharmonic solution to the equation p u = ω in R n such that lim inf x u = 0, then u W,p loc (Rn ) L loc (Rn, dω). Proof. Let us first show that u L loc (Rn, dω) using Wolff s inequality [HW83]. Fix a ball B = B(0, R), R > 0. By Theorem 2.6, u satisfies the Wolff potential estimate 26

34 (2.2). Hence, B u dω K + K ( R ω(b(x, r)) B 0 B R r n p ( ω(b(x, r)) r n p dr r dω(x) dr dω(x) := I + II. r Since B(x, r) 2B = B(0, 2R) for x B and r < R, we obtain by (2.2), R ( ω(b(x, r) 2B) I K B 0 r n p K W,p ω 2B dω 2B <. R n dr r dω(x) To estimate II, notice that B(x, r) B(0, 2r), for r > R and x B. Hence, II Kω(B) R ( ω(b(0, 2r)) r n p dr r < by Corollary 2.0. We next show that u L s loc (Rn, dx) for 0 < s np. Arguing as above, we use n p (2.2) and split the integral with respect to dr/r into two parts: B u s dx c s K s + c s K s B B ( R ( ω(b(x, r)) 0 r n p ( ( ω(b(x, r)) R r n p ) s dr dx r dr r ) s dx := III + IV, where c is a constant depending only on s. To estimate III, notice that by (2.) ω 2B L,p (R n ), and consequently there is a unique solution u 2B L,p 0 (R n ) to the equation p u 2B = ω 2B in R n. Hence, by the 27

35 Sobolev inequality, u 2B L s loc (Rn ) for 0 < s np n p. Clearly, u 2B is p-superharmonic, and satisfies (2.2) with ω 2B in place of ω, i.e., 0 ( ω(b(x, r) 2B) r n p dr r K u 2B(x). Since B(x, r) 2B for x B and r < R, we estimate ( R III c B 0 ( ω(b(x, r) 2B) r n p ) s dr dx c u s r 2Bdx <. B The estimate of IV is similar to that of II: ( ( ω(b(0, 2r)) IV c s K B R r n p ) s dr < r by Corollary 2.0. Thus, u L s loc (Rn, dx) for s np n p. We next show that there exists 0 < β such that, for all balls B, (2.23) B Du p u β dx <. Indeed, since u is p-superharmonic, it is a locally renormalized solution to p u = ω as discussed in Chapter 2. Let u k = min (u, k), where k > 0. Note that u, and hence u k, is locally bounded below. Using h(u) = u β k (0 < β ) in (2.9), and a cut-off function ϕ C0 (B) such that 0 ϕ and ϕ = on B, we obtain 2 (2.24) β u k Du p u β ϕ dx + B Du p 2 Du ϕ u β k dx = B u β k ϕ dω. As was shown above, u L loc (Rn, dω), and hence the right-hand side is bounded 28

36 by (2.25) ( β u β ϕ dω ω(b β u dω) <, B B for 0 < β. Since u is p-superharmonic, we have Du L r () for r < inequality with exponents r and r > n, we deduce from (2.24), n. By Hölder s n (2.26) β u k Du p u β ϕ dx c Du L r () (B,dx) u β L βr (B,dx) ( β + ω(b β u dω). B If βr = s np, where r > n and β, then the right-hand side of the preceding n p inequality is finite. Picking r so that r > n and is arbitrarily close to n, and passing to the limit as k, we obtain (2.23) for β = β 0, provided 0 < β 0 < p n p, β 0. In the case p n p >, i.e., for p > n 2, we can set β 0 =, which shows that in fact Du L p ( B, dx), for all B = B(0, R). Hence, Du = u in the distributional sense, 2 and consequently u W,p loc (Rn ). For < p n np, we fix s so that p < s which ensures that u 2 n p Ls loc (Rn, dx) as shown above. Applying Hölder s inequality with exponents p and p, we obtain 29

37 from (2.24) and (2.25), β u k ( ( ) Du p u β ϕ dx c Du p u β0 p dx u βp+( β0)() p dx B B ( β + ω(b β u dω). Passing to the limit as k, we deduce that (2.23) holds if β and B βp + ( β 0 )(p ) βp + p s. ( In particular, (2.23) holds for β = β = min, s p+ p If β =, then u W,p loc (Rn ) as above. In the case ). β = s p + p <, we set β j = β + β p j, so that β j p + ( β j )(p ) = s, j 2. In other words, Since β j = s p + p j ( ) i p, j =, 2,.... p i=0 lim β j = s p + >, j we can choose J 2 so that β β J <, but β J. 30

38 If β J >, then we will replace β J with β J =. Clearly, β j p + ( β j )(p ) = s, j = 2, 3,..., J ; β J p + ( β J )(p ) s. Arguing by induction, and using (2.23) with β = β j, for j = 2, 3,..., J, we estimate as above, β j u k ( B ( Du p u βj ϕ dx c B Du p u βj p dx ( u β jp+( β j )() p dx + ω(b) β j Since β J = at the last step, we arrive at the estimate u k Du p ϕ dx C B <, B u dω) βj <. where C B does not depend on k. Passing to the limit as k, we conclude that u W,p loc (Rn ). (2.). In the next theorem we obtain a lower bound for solutions of the integral inequality Theorem 2.2. Let < p < n, 0 < q < p, 0 < α < n p, and σ M + (R n ). Suppose 0 u L q loc (Rn, dσ) is a nontrivial solution to (2.). Then there holds (2.27) u(x) C [W α,p σ(x)] q, x R n, where C is a positive constant depending only on n, p, q, and α. 3

39 The same lower bound holds for a nontrivial p-superharmonic supersolution to (.). If p n, there is only a trivial supersolution u = 0 on R n. Before proving Theorem 2.2, we need the following lemma. Lemma 2.3. Let < p < and 0 < α < n. Then, for every r > 0, p (2.28) W α,p [(W α,p σ) r dσ] c r (Wα,p σ) r +, where c = c n,p,α > 0 depends only on n, p, and α. Proof. For t > 0, obviously, W α,p σ(y) = t 0 ( σ(b(y, s)) s n αp ds s + t ( σ(b(y, s)) s n αp ds s. For y B(x, t), we have t ( σ(b(y, s)) s n αp ds s = t/2 ( σ(b(y, 2r)) (2r) n αp dr r = ( ) n αp 2 t/2 ( σ(b(y, 2s)) s n αp ds s C n,p,α t ( σ(b(y, 2s)) s n αp where C n,p,α = ( ) n αp. Since s t and y B(x, t), then B(y, 2s) B(x, s), which 2 implies ( σ(b(x, s)) (2.29) W α,p σ(y) C n,p,α t s n αp ds s. ds s, 32

40 Notice that W α,p ((W α,p σ) r dσ)(x) = 0 ( B(x,t) [W α,pσ(y)] r t n αp dt t. By (2.29), we obtain 0 B(x,t) [ W α,p ((W α,p σ) r dσ)(x) C n,p,α t ( σ(b(x,s)) s n αp t n αp ds s ] r dt t C r n,p,α Integrating by parts, we deduce Clearly, 0 [ ( σ(b(x, s)) W α,p ((W α,p σ) r dσ)(x) t s n αp C r n,p,α r + ds ] r s ( σ(b(x, t)) t n αp ( ( σ(b(x, s)) 0 s n αp dt t. ) r + ds. s r + e r, and hence, (2.28) follows with c = e C n,p,α. This completes the proof of Lemma 2.3. In particular, setting α = and r = q() in Lemma 2.3, we deduce the following q estimate used extensively below: (2.30) W,p ( (W,p σ) q() q dσ ) (x) κ (W,p σ(x)) q, where κ depends only on p, q, and n. Proof of Theorem 2.2. Let dω = u q dσ. Fix x R n and pick R > x. Let B = 33

41 B(0, R), and let dσ B = χ B dσ. Iterating (2.), we obtain We estimate, u(x) W α,p [(W α,p ω) q dσ B ] (x) ( = W t n αp α,p ω(z) q dσ(z) 0 B(x,t) B dt t. W α,p ω(z) = 0 ( ω(b(z, s)) s n αp ds s c R ( ω(b(z, 2s)) s n αp ds s, where c = c(n, p, α) > 0. Notice that if z B and R s then B(z, 2s) B(0, s). Hence, W α,p ω(z) c R ( ω(b(0, s) s n αp ds s. From this it follows, u(x) [c M(R)] q Wα,p σ B (x), where M(R) = R ( ω(b(0, s) s n αp ds s > 0. Combining (2.) with the preceding estimate, and using Lemma 2.3 with r = q and σ B in place of σ, we obtain q u(x) [c M(R)] ( )2 W α,p [(W α,p σ B ) q dσ] (x) c q [c M(R)] ( q )2 [W α,p σ B (x)] + q. 34

42 Iterating this procedure and using Lemma 2.3 with r = q j k=0 ( q )k, we deduce u(x) c j k( q k= ) k [c M(R)] ( q )j+ [W α,p σ B (x)] j k=0 ( q )k, for all j = 2, 3,.... Since 0 < q < p, obviously k= ( ) k q k <. p Letting j in the preceding estimate we obtain u(x) C [W α,p σ B (x)] q, B = B(0, R), R > x, where C > 0 depends only on n, p, q, and α. Letting R yields (2.27) for all x R n. The next lemma shows that if there exists a nontrivial solution to (2.), then σ must be absolutely continuous with respect to the (α, p)-capacity defined by (2.4) (see [AH96], Sec. 2.2). As a consequence, if (.) has a nontrivial p-superharmonic supersolution, then σ is absolutely continuous with respect to the p-capacity defined by (.25). Notice that cap p (E) cap,p (E) for compact sets E. Lemma 2.4. Let < p <, 0 < q < p, 0 < α < n p, and σ M + (R n ). Suppose there is a nontrivial solution u L q loc (Rn, dσ) to inequality (2.). Then there exists a constant C depending only on n, p, q, α such that (2.3) σ(e) C [ cap α,p (E) ] ( q E ) q u q dσ, 35

43 for all compact sets E R n. Proof. Let dω = u q dσ. Then u W α,p ω dσ-a.e. By Theorem. in [V], E dω (W α,p ω) C cap α,p(e), where C depends only on n, p, and α. Hence, (2.32) E u q p+ dσ E dω (W α,p ω) C cap α,p(e). Note that q p + < 0. Using Hölder s inequality with exponents r = q r =, we have q and where β = q( q) σ(e) = E ( u β u β dσ the preceding estimate implies (2.3). E ( u βr r dσ u βr r dσ, E > 0. Then βr = q p+ and βr = q, and since u L q loc (Rn, dσ), 36

44 Chapter 3 Finite energy solutions 3. Main results In this chapter, we study finite energy solutions to quasilinear elliptic equation (.). We are interested in solutions u L,p 0 (R n ) to (.), and related integral inequalities. Here L,p 0 (R n ) is the homogeneous Sobolev (or Dirichlet) space defined in Chapter 2 (see [HKM06], [MZ97], [Maz]). More precisely, u is called a finite energy solution to (.) if u L,p 0 (R n ) L q loc (Rn, dσ), u 0, and, for all ϕ C 0 (R n ), (3.) u p 2 u ϕ dx = u q ϕ dσ. R n R n Finite energy solutions to (.) are critical points of the functional H[ϕ] = R p ϕ p dx n R q + ϕ +q dσ. n 37

45 We will give a necessary and sufficient condition for the existence of a nontrivial finite energy solution to (.), and prove its uniqueness. Our main result is the following. Theorem 3.. Let 0 < q < p, < p < n, and let σ M + (R n ). Then there exists a nontrivial solution u L,p 0 (R n ) L q loc (Rn, dσ) to (.) if and only if (3.2) R n (W,p σ) (+q)() q dσ <. Furthermore, such a solution is unique. For p n, (.) has only a trivial solution u = 0. We observe that (3.2) yields σ L,p loc (R n ), where L,p (R n ) = L,p 0 (R n ) is the dual Sobolev space (see definitions in Chapter 2). absolutely continuous with respect to the p-capacity cap p ( ). Consequently, σ is necessarily Moreover, as was shown in [COV00], condition (3.2) holds if and only if there exists a constant C such that, for all ϕ C 0 (R n ), (3.3) ( R n ϕ +q dσ +q C ϕ L p (R n ). An obvious sufficient condition which follows from Sobolev s inequality is σ L r (R n ), r = np n( q)+p(+q). There is also an equivalent characterization of (3.3) in terms of capacities due to Maz ya and Netrusov (see [Maz], Sec..6): (3.4) σ(r n ) 0 ( +q t q dt < +, κ(σ, t) 38

46 where κ(σ, t) = inf{ cap p (E) : σ(e) t}. Thus, any one of the conditions (3.2), (3.3), and (3.4) is necessary and sufficient for the existence of a nontrivial finite energy solution to (.). We now outline the contents of this chapter. In Sec. 3.2 we study the corresponding integral inequalities, deduce a necessary and sufficient condition for the existence of a finite energy solution, and construct a minimal solution. In Sec. 3.3 we prove the uniqueness property of finite energy solutions. 3.2 Existence and minimality of finite energy solutions In this section, we deduce a necessary and sufficient condition for the existence of a finite energy solution, and construct a minimal solution to (.). We will assume that < p < n, since for p n there are only trivial nonnegative supersolutions on R n (Theorem 2.2; see also [HKM06], Theorem 3.53). We start with the following lemma. Lemma 3.2. Suppose there exists a nontrivial supersolution u 0, u L,p 0 (R n ) L q loc (Rn, dσ) to (.). Then p u L,p (R n ) M + (R n ). Moreover, u L +q (R n, dσ) (for a quasicontinuous representative of u), and condition (3.2) holds. Proof. Suppose u L,p 0 (R n ) L q loc (Rn, dσ) is a supersolution to (.). By Hölder s 39

47 inequality, we have, for every ϕ C0 (R n ), p u, ϕ = u p 2 u ϕ dx u L p (R n ) ϕ L p (R n ). R n Hence, p u L,p (R n ). If ϕ 0, then p u, ϕ = u p 2 u ϕ dx R n ϕ u q dσ 0, R n 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 sequence of nonnegative C 0 -functions such that ϕ j u in L,p 0 (R n ). By definition, R n u p 2 u ϕ j dx µ, ϕ j. Hence, u p dx = lim u p 2 u ϕ j dx lim µ, ϕ j = µ, u. R n j R n j Let us assume as usual that u coincides with its quasicontinuous representative. Then, applying Theorem 2., we deduce µ, u = u dµ = R n u +q dσ <. R n By Theorem 2.2, it follows that if u 0, then u C ( W,p σ ) q, and consequently (3.2) holds. 40

48 Let us define a nonlinear integral operator T by (3.5) T (f)(x) = ( W α,p ( f dσ)) (x), x R n. Lemma 3.3. Let < p <, 0 < q < p, 0 < α < n p and β > 0. Let σ M + (R n ). Suppose (3.6) R n (W α,p σ) (β+q)() q dσ <. Then T is a bounded operator from L β+q q (R n, dσ) to L β+q (R n, dσ). Proof. Let f L β+q q (R n, dσ). Without lost of generality, we may assume that f 0. Clearly, (W α,p (fdσ)) L β+q (dσ) = (R n ( ) ) β+qdσ β+q W α,p (fdσ). We have W α,p (fdσ)(x) 0 ( σ(b(x, r)) r n αp ) p M σ f(x) p dr r = M σf(x) p W α,p σ(x), where the centered maximal operator M σ is defined by M σ f(x) = sup r>0 f dσ, x R n. σ(b(x, r)) B(x,r) It is well known that M σ : L s (R n, dσ) L s (R n, dσ) is a bounded operator for all s >. Let s = β+q q. Then, using Hölder s inequality with the exponents q > and 4

49 , we estimate, q Thus, R n ( ( C ( ) β+qdσ Rn W α,p (fdσ) (M σ f) β+q (Wα,p σ) β+q dσ (M σ f) β+q q dσ R n f β+q q dσ R n ) q ( Rn(W α,p σ) (β+q)() q ) q ( Rn(W α,p σ) (β+q)() q W α,p (fdσ) L β+q (dσ) c f L β+q q (dσ). ) q dσ ) q dσ. Remark 3.4. It is not difficult to see that actually (3.6) is also necessary for the boundedness of the operator T : L β+q q (R n, dσ) L β+q (R n, dσ) (see, for example, [COV06]). Theorem 3.5. Let < p < n, 0 < q < p, 0 < α < n, β > 0 and σ M + (R n ). p Suppose that condition (3.6) holds. Then there exists a solution u L β+q (R n, dσ) to the integral equation (.8). Conversely, (3.6) is also necessary for the existence of a solution u L β+q (R n, dσ) to (.8). Proof. Suppose that (3.6) holds. By Lemma 3.3, we have, for all nonnegative f L β+q q (R n, dσ), (3.7) R n ( ) (Rn ) q β+q W α,p (fdσ) dσ C f β+q q dσ. Let u 0 = c 0 (W α,p σ) q, where c0 > 0 is a small constant to be chosen later on. We 42

50 construct a sequence of iterations u j as follows: (3.8) u j+ = W α,p (u q jdσ), j = 0,, 2,.... Applying Lemma 2.3, we have q u = W α,p (u q 0dσ) = c0 W α,p ((W α,p σ) q() q q dσ) c q 0 c q (Wα,p σ) q, where c is the constant in (2.28). Choosing c 0 so that c q q 0 c q c 0, we obtain u u 0. By induction, we can show that the sequence {u j } j is nondecreasing. Note that u 0 L β+q (R n, dσ) by assumption. Suppose that u 0,..., u j L β+q (R n, dσ) for some j 0. Then R n u β+q j+ dσ = R n ( Wα,p (u q j dσ)) β+q dσ. Applying (3.7) with f = u q j, we obtain by induction, (3.9) R n u β+q j+ dσ C ( u β+q j R n ) q dσ <. Since u j u j+, the preceding inequality yields ( ) q u β+q j+ dσ C u β+q j+ dσ <. R n R n Thus, ( ) q u β+q j+ dσ R n C <. Using the Monotone Convergence Theorem and passing to the limit as j in (3.8), we see that there exists u = lim j u j, such that u L β+q (R n, dσ) and u is a 43

51 nontrivial solution to (.8). Conversely, suppose that there exists a solution u L β+q (R n, dσ) to (.8). By Theorem 2.2, u C(W α,p σ) q, and hence (3.6) follows. Lemma 3.6. Let u L +q (R n, dσ) be a nonnegative solution to the integral inequality (2.) with α =. Then (3.0) u q dσ L,p (R n ). Proof. Let dν = u q dσ. We need to show that, for all ϕ C 0 (R n ), (3.) ϕ dν c ϕ R (R p dx n n p. It is easy to see that the above inequality is equivalent to (3.2) I g dν c g R (R p dx n n p, for all g L p (R n ), where I g is the Riesz potential of g of order. By duality, (3.2) is equivalent to (3.3) (I ν) p dx <. R n Using Wolff s inequality (2.8), we deduce that (3.3) holds if and only if (3.4) R n W,p ν dν <. 44

52 Notice that since u W,p (u q dσ) and u L +q (R n, dσ) then W,p ν dν = R n W,p (u q dσ) u q dσ R n u +q dσ <. R n Thus, (3.3) holds. This completes the proof of the lemma. We will need a weak comparison principle which goes back to P. Tolksdorf s work on quasilinear equations. The following lemma is essentially known; we include a proof in the context of quasicontinuous solutions on the entire space for the convenience of the reader (see also [PV08], Lemma 6.9, for 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 0 (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 0 (R n ), we have by Theorem 2., (3.5) u p 2 u ϕ dx = µ, ϕ = ϕ dµ, R n R n (3.6) v p 2 v ϕ dx = ω, ϕ = ϕ dω. R n R n Hence, (3.7) ( u p 2 u v p 2 v) ϕ dx = R n ϕ dµ R n ϕ dω. R n 45

53 Since µ ω, it follows that, for every ϕ L,p 0 (R n ), ϕ 0, we have (3.8) R n ( u p 2 u v p 2 v) ϕ dx 0. Testing (3.8) with ϕ = (u v) + = max{u v, 0} L,p 0 (R n ), we obtain, I = ( u p 2 u v p 2 v) (u v) + dx 0. R n Let A = {x R n : u(x) > v(x)}, then I = ( u p 2 u v p 2 v) (u v) dx 0. A Note that ( u p 2 u v p 2 v) (u v) 0. Thus, 0 ( u p 2 u v p 2 v) (u v) dx = ϕ(dµ dω) 0. A A 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} (u v) p dx = 0. a p A Consequently, cap p (A) = 0, i.e., u v q.e. We are now in a position to prove the main theorem of this section. Theorem 3.8. Let < p < n and 0 < q < p. Let σ M + (R n ), σ 0. Suppose 46

54 that (3.2) holds. Then there exists a nontrivial solution w L,p 0 (R n ) L q loc (Rn, dσ) to (.). Moreover, w is a minimal solution, i.e., w u dσ-a.e. (q.e. for quasicontinuous representatives) for any nontrivial solution u L,p 0 (R n ) L q loc (Rn, dσ) to (.). Proof. We first show that there exists a solution w L,p 0 (R n ) L q loc (Rn, dσ) to (.). Since (3.2) holds, applying Theorem 3.5 with α = and β =, we conclude that there exists a solution v L +q (R n, dσ) to the integral equation (.8) with α =. By using a constant multiple c v in place of v, we can assume that v = KW,p (v q dσ), where K is the constant in (2.2). Then by Lemma 3.6 and Theorem 2.2, v q dσ L,p (R n ), and v C K q (W,p σ) q, where C is the constant in (2.27). We set w 0 = c 0 (W,p σ) q, dω0 = w q 0 dσ, where c 0 > 0 is a small constant to be determined later. We see that w 0 v if c 0 CK q. Hence, w 0 L +q (R n, dσ), and ω 0 L,p (R n ). Then there exists a unique nonnegative solution w L,p 0 (R n ) to the equation p w = ω 0, and w,p = ω 0,p. (See (2.4)). Moreover, by Theorem 2.6, 0 w K W,p ω 0 K W,p (v q dσ) = v. 47

55 Consequently, by Lemma 3.6, w L +q (R n, dσ), and w q dσ L,p (R n ). We deduce using (2.30), q w K W,pω 0 = c ( ) 0 K W,p (W,p σ) q() q dσ q 0 c q c q K (W,p σ) q = c q q 0 c q K w 0. Hence, for c 0 (K c q q ) q, we have v w w 0. To prove the minimality of w, we will need c 0 C, so we pick c 0 such that (3.9) 0 < c 0 min { C K q, (K c q q ) q, C }. Let us now construct by induction a sequence {w j } j so that (3.20) p w j = σ w q j in Rn, w j L,p 0 (R n ) L +q (R n, dσ), 0 w j w j v, q.e., w q j dσ L,p (R n ), where sup j w j,p <. We set dω j = w q j dσ, so that p w j = ω j, j =, 2,.... Suppose that w 0, w,..., w j have been constructed. As in the case j =, we see that, since ω j L,p (R n ), there exists a unique w j L,p 0 (R n ) such that 48