REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents

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1 REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM BERARDINO SCIUNZI Abstract. We prove some sharp estimates on the summability properties of the second derivatives of solutions to the equation p u = f(x), under suitable assumptions on the source term. As an application we deduce some strong comparison principles for the p-laplacian, in the case of vanishing source terms. Contents. Introduction 2. Local regularity 7 3. Local summability of the weight 2 4. The strong comparison principle 7 5. Examples 9 References 20. Introduction In this paper we study the regularity of weak solutions to (.) p u = f(x) in a domain R n, n 2. The issue of regularity, as we will see, is also related to the validity of the strong comparison principle. A solution u to (.) can be defined e.g. assuming that u W,p () in the weak distributional meaning. This is also the space where it is natural to prove the existence of the solutions under suitable assumptions. Nevertheless, under our assumptions on the source term f, it follows Date: November 23, Mathematics Subject Classification. 35B5,35J92,35B65. Key words and phrases. p-laplace equations, regularity theory, Harnack inequality, comparison principles.

2 2 BERARDINO SCIUNZI by [8, 29] that u C,α () for some 0 < α <, see also [4] for the regularity up to the boundary. On the contrary, solutions to p-laplace equations generally are not of class C 2 (). We recall in Section 5 two leading examples (Example 5. and Example 5.2) for the reader s convenience, that show the existence of solutions that are not in C 2 (). Nevertheless it is possible to show that, in the case p > 2 and strictly positive (or negative) source terms, any solution is not of class C 2 at its critical points. We add the details of this fact in Proposition 5.4. Therefore two crucial issues arises from this fact: - the study of integrability properties of the second derivatives of the solutions, - the study of the maximal exponent for the Hölder continuity of the gradients. One of the first regularity results regarding the second derivatives of the solutions states that, under suitable assumptions on f, we have: u p 2 u W,2 loc (). This result can be found in [5] for the general case f L s () with s > max{2, n }. Let us mention that this is also implicit in [8, 3, 29]. We p also refer the readers to [24] for W 2,2 estimates in the case of singular ( < p < 2) quasilinear elliptic equations involving singular potentials. In [5, 25] it has been shown that: if u C () is a solution to (.), then i) u W 2,2 loc ii) u W 2,q loc away from zero. () if < p < 3. p () for any q < p 2 if p 3 and if f is strictly bounded Actually in [5] positive solutions to p u = f(u) were considered. Anyway the arguments of [5] apply also in our context providing i) and ii) that are in any case a consequence of the results in this paper. Note that the degenerate nonlinear nature of the p-laplacian causes that the Calderón-Zygmund theory can not be extended simply to the case of p-laplace equations. We refer the readers to [7, 8] for a very interesting extension of the Calderón-Zygmund theory to the quasilinear case. It is worth emphasizing that, as it follows observing the -D solution in Example 5.2, the exponent q in ii) here above is optimal. On the contrary the radial solutions in Example 5. are more regular. This phenomenon (see Remark 5.3) highlights the important role played by the critical set Z u of the solution u: (.2) Z u := { u = 0}.

3 REGULARITY AND COMPARISON PRINCIPLES 3 The critical set Z u is in fact the set of points where the p-laplace operator is singular ( < p < 2) or degenerate (p > 2). Away from the critical set standard regularity theory applies. It is an open problem to understand if the positivity assumption on the source term in ii) can be removed. We will show here that in general it is not necessary. One of the purposes of this paper is to point out the strong relation between the study of the integrability properties of the second derivatives of the solutions and the study of the maximal exponent for the Hölder continuity of the gradient, starting from a recent important result of Eduardo V. Teixeira [27] which is based on previous estimates obtained in []. Namely, exploiting the results in [] and also some techniques from [], it has been shown in [27] that, if u W,p () is a solution to (.) and (.3) f L s () with s > n, then (.4) u C,min{α M, s n (p )s }, that is u C,α s n for any α (0, α M ) (0, ] where α (p )s M is the maximal exponent for the C,α regularity of p-harmonic functions. We also refer the interested readers to [9,, 9, 20, 28]. Let us point out that (.3) is in particular implied by stronger assumptions that we will consider in our results. It is in any case important to state explicitly (.3) since the parameter s will appear in our statements and in some cases it could be different by the one obtained by embedding theorems. Let us also mention that in [27] more general problems are considered including in particular operators with variable coefficients. Improving the technique in [5] and exploiting (.4), we get some weighted estimates that in particular show that the positivity of f is not necessary to get ii). Let us set (.5) 0 < µ := min{α M, s n }(p ), (p )s with s > n as in (.3). Also set (.6) 0 < µ := min{α M (p ), }, that is the limiting case for s. We have the following: Theorem.. Let u W,p () be a solution to (.). Assume that (.3) holds and f W,n (). Consider a critical point x 0 Z u with. Then: if < p < 3

4 4 BERARDINO SCIUNZI we have that (.7) B ρ(x 0 ) D 2 u 2 x x 0 η C with C = C(p, n, f, u, x 0 ) and for any η such that η < η := n ( 2 min{α M, s n (p )s } 2), with s > n given by (.3). In particular u W 2,2 loc (). If else p 3 and f satisfies (I µ ) (see (.2) and (.3)), then, for any q < p, we p 2 have: D 2 u q (.8) x x 0 τ C, for any B ρ(x 0 ) (.9) τ < τ := n 2 q 2 µ, with µ defined in (.5). Theorem. is a consequence of Corollary 2.2 and Corollary 3.2 that improves the results in [5]. In fact it is not required the strict positivity of f as in [5] and the exponents to the singular weights in (.7) and (.8) are optimal, as it can be deduced from the examples in Section 5. To guess this, the reader should evaluate the integral in (.8) in the case of the solution of Example 5.2. As an application it is interesting to state Theorem. for the case of Hénon-type equations: (.0) p u = x σ g(u), σ 0. We have the following: Theorem.2. Let u C () be a solution to (.0) with 0 and 0 σ < µ. Let x 0 Z u with. Then, if < p < 3 we have that (.7) holds for u for any η such that η < η := n ( 2 min{α M, (p ) } 2). In particular u W 2,2 p loc (). If else p 3 then, for any q <, we p 2 have that (.8) holds true for u for any τ such that: (.) τ < τ := n 2 q 2 µ, with µ defined in (.5).

5 REGULARITY AND COMPARISON PRINCIPLES 5 Theorem.2 follows by applying Theorem. with f(x) := x σ g(u(x)) L (). The regularity theory developed in Section 2 is very much related to study of the summability properties of u. This is a consequence of regularity estimates at first and later it allows to improve the regularity results, as it is the case of Corollary 3.2. The summability of u near the critical set Z u is in some sense a measure of the degeneracy of our equation at its critical points. This is an information that allows to get weighted Sobolev inequalities and that consequently turns out to be crucial in many applications. This is the case in particular when dealing with the strong comparison principle for p-laplacian. Consider a differential operator L : D R (to be understood in the weak sense if necessary), defined on its domain D = D() (a function space over some domain ). We say that the strong comparison principle holds for L in if, for any (connected) subdomain and any u, v D such that Lu Lv, then the assumption u v in implies the alternative u < v or u v in. The semilinear case is well understood and we refer the readers to [0]. Note that, for example, in the case Lu := u g(x, u) with g locally Lipschitz continuous w.r.t. the second variable, the strong comparison principle reduces to the maximum principle. It is crucial here the fact that the Laplace operator is linear. Nevertheless, also in the quasilinear case, the situation is well understood far away from the critical set Z u. Namely, in the case Lu := p u g(x, u) and p > with g locally Lipschitz continuous w.r.t. the second variable, the strong comparison principle holds in any connected component of \ Z u (see [3, 23]). Near the critical set, the nonlinear nature of the p-laplacian is in addition to the degeneracy of the operator and very few is known, even in the p-harmonic case Lu := p u and assuming furthermore that both u and v are p-harmonic. Let us mention [2] for some results in the p-harmonic case in dimension two. In the general case Lu := p u g(x, u) it has been proved in [6] that the strong comparison principle holds (also over the critical set) provided that u or v are a solution of the equation, 2n2 < p < and the n2 right hand side g(x, u) is strictly positive.

6 6 BERARDINO SCIUNZI It is a common feeling that, at least in the case p > 2, the strong comparison principle could fail if the source term change sign. However, as we will see here, the strict positivity of the source term is not necessary. Namely we will provide some conditions on the (possibly vanishing) source term under which the strong comparison principle holds. We will in particular assume that f satisfies the condition (I µ ) here below. Namely, for µ defined as in (.5) we state: Condition (I µ ): for some 0 < µ < µ, we have: (.2) f W,m n () for some m > 2( µ) and, given any x 0, there exist C(x 0, µ) > 0 and ρ(x 0, µ) > 0 such that, B ρ(x0,µ)(x 0 ) and (.3) f(x) C(x 0, µ) x x 0 µ in B ρ(x0,µ)(x 0 ). We have the following: Theorem.3. Let u, v C () such that (.4) p u f(x) p v f(x) in. Assume that u or v is a solution to (.) with 2n2 n2 < p < and assume that f satisfies (.3) and (I µ ). Then, if u v in a (connected) subdomain, it follows that u < v in, unless u v in. Let us point out that, even if f may be deleted in (.4), we need in any case that one of the functions we are comparing is a solution to the equation. As an example, we state here below a strong comparison principle for Hénon type problems. We have the following: Theorem.4. Let u, v C () such that (.5) p u x σ g(u) p v x σ g(v) in, with 0, 0 σ < µ (see (.6)) and g( ) nonnegative locally Lipschitz continuous with g(s) > 0 for s > 0. Assume that u or v is a (nontrivial) non-negative solution to (.0) with 2n2 < p <. n2 Then, if u v in a (connected) subdomain, it follows that u < v in, unless u v in. Theorem.3 and Theorem.4 follow by a Harnack-type comparison inequality, see Theorem 4.2, that can be proved exploiting the iterative technique in [30] which goes back to [7, 2, 22] and was first used to prove Hölder continuity properties of solutions of some strictly elliptic linear operators. To apply this technique in our case the key tool is

7 REGULARITY AND COMPARISON PRINCIPLES 7 the weighted Sobolev inequality in Theorem 4. that can be obtained once the regularity results of Theorem. are available. The paper is organized as follows. In Section 2 we prove some weighted regularity estimates for the solutions to (.), that we exploit in Section 3 to prove summability properties of u avoiding the assumption that f is strictly bounded away from zero needed in [5]. In Section 4 we prove Theorem.3 and Theorem.4. Finally in Section 5 we collect some examples showing which is the best regularity expected for the solutions to (.). 2. Local regularity Let u W,p () be a solutions to (.), that is (2.6) u p 2 ( u, φ) = f(x)φ for every φ W,p 0 (). Note that actually u C,α () under our assumptions. We will get our regularity results exploiting the linearized operator at u. To define it let us start recalling some known facts about weighted Sobolev spaces. It is natural to distinguish the case < p < 2 from the case p 2. The case p 2. The weighted Sobolev space (with weight ρ) W,2 (, ρ) can be defined as the set of those functions having distributional derivative for which the norm: (2.7) v ρ = v L 2 () v L 2 (,ρ) is bounded. We are interested in particular to the case ρ = u p 2 which is the weight that is naturally associated to our problem. Furthermore we can also define the weighted Sobolev space H,2 (, ρ) as the closure of C () in W,2 (, ρ) w.r.t. the norm in (2.7). Note that, since p 2, then ρ L () so that C () W,2 (, ρ) and furthermore it follows that H,2 (, ρ) W,2 (, ρ). A posteriori, according to Theorem 3. (see also Theorem 4.), we have that ρ L (). This allows to exploit the result of Meyers and Serrin [6], as remarked in [30] and to deduce that in any domain actually H,2 (, ρ) = W,2 (, ρ). Note now that H,2 0 (, ρ) may be defined as the closure of Cc () in W,2 (, ρ) w.r.t. the norm in (2.7). H,2 0 (, ρ) also coincides with the completion of Cc () w.r.t. the norm in (2.7). This will be a consequence of the regularity results (namely Theorem 3.) that we are going to prove. To do this, at the beginning, we have to choose a space where to define the linearized operator. We define (in the case p 2) the linearized operator in the space H,2 (, ρ) and,

8 8 BERARDINO SCIUNZI for any v H,2 (, ρ), we consider the linearized equation L u (v, ) = 0 associated to (.) at a given solution u defined by (2.8) L u (v, ϕ) = u p 2 ( v, ϕ) (p 2) u p 4 ( u, v)( u, ϕ) for every ϕ H,2 0 (, ρ), where we have posed: f i (x) = f x i (x). It follows by [5] that, setting u i (x) = u x i (x), then u i H,2 (, ρ) and (2.9) L u (u i, ϕ) = 0 ϕ H,2 0 (, ρ). The case < p < 2. In this case, a posteriori, according to Theorem 3. (see also Theorem 4.), the above construction can be carried out as well, since we will prove that ρ L (). At the beginning, to achieve Theorem 3., we start defining the linearized operator in the space A u := { v H 0() : v L 2 (,ρ 2 ) < }. It follows by [5] that u i A u and (2.20) L u (u i, ϕ) = 0 ϕ H,2 0 (). We have the following: Theorem 2.. Let u W,p () be a solution to (.), < p <. Assume that (.3) holds and let x 0 Z u with. Fix 0 β <, 0 µ < µ and γ < n 2 for n 3 while γ = 0 if n = 2. Then, if f W,m () for some m > (2.2) B ρ(x 0 ) n 2 µ, it follows that: u p 2 β u i,j 2 C i, j {,..., n}, x x 0 µ x y γ with C = C(x 0, ρ, f, n, p, β, γ, µ) that does not depend on y. Proof. We start recalling that, since we assumed that (.3) is satisfied, then the results in [27] apply and (.4) holds. Let us fix some notations. We consider G ε (t) = (2t 2ε)χ [ε, 2ε] (t) tχ [2ε, ) (t) for t 0, while G ε (t) = G ε ( t) for t 0 (χ [a,b] ( ) denoting the characteristic function of a set). Consider a cut-off function φ ρ, in such a way that φ ρ Cc (), φ ρ = in B ρ (x 0 ) and φ ρ 2. For shortness we will write φ instead ρ of φ ρ. Also, for 0 β <, γ < (n 2) if n 3, γ = 0 if n = 2 and µ < µ fixed, we set (2.22) T ε (t) = G ε(t) t β and H δ (t) = G δ(t) t γ, f i ϕ

9 REGULARITY AND COMPARISON PRINCIPLES 9 and consider the test function ϕ = T ε (u i ) Hδ( x y ) x x 0 µ φ 2. Note that ϕ can be plugged into (2.9) (see also (2.20)) since it is regularized near the critical set Z u by the definition of T ε (note also that x 0 Z u ). Moreover ϕ is regularized near any y because of the definition of H δ. Finally ϕ is smooth elsewhere by standard regularity theory. Therefore we have (2.23) u p 2 u i 2 T ε(u i ) Hδ( x y ) φ 2 x x 0 µ (p 2) u p 4 ( u, u i ) 2 T ε(u i ) Hδ( x y ) φ 2 x x 0 µ u p 2 φ 2 ( u i, x H δ ( x y )) T ε (u i ) x x 0 µ (p 2) u p 4 φ 2 ( u, u i )( u, x H δ ( x y )) T ε (u i ) x x 0 µ µ u p 2 ( u i, x x 0 ) T ε (u i ) Hδ( x y ) φ2 x x 0 µ µ(p 2) u p 4 ( u, u i )( u, x x 0 ) T ε (u i ) Hδ( x y ) φ2 x x 0 µ 2 u p 2 ( u i, φ) T ε (u i ) Hδ( x y ) φ x x 0 µ 2(p 2) u p 4 ( u, u i )( u, φ) T ε (u i ) Hδ( x y ) φ x x 0 µ = f i (x) T ε (u i ) Hδ( x y ) φ 2. x x 0 µ The main idea in the following computations is to estimate all the terms in the previous inequality and then close the estimates via Hölder s inequality. We start observing that min{(p ), } u p 2 u i 2 T ε(u i ) Hδ( x y ) x x 0 µ φ 2 u p 2 u i 2 T ε(u i ) Hδ( x y ) x x 0 µ φ 2 (p 2) u p 4 ( u, u i ) 2 T ε(u i ) Hδ( x y ) x x 0 µ φ 2

10 0 BERARDINO SCIUNZI and get by (2.23) that C u p 2 u i 2 T ε(u i ) Hδ( x y ) φ 2 x x 0 µ u p 2 φ 2 u i x H δ ( x y ) T ε (u i ) x x 0 µ u p 2 u i T ε (u i ) Hδ( x y ) (2.24) φ2 x x 0 µ u p 2 u i φ T ε (u i ) Hδ( x y ) φ x x 0 µ f i (x) T ε (u i ) Hδ( x y ) φ 2. x x 0 µ Here and in the following we denote with C = C(x 0, ρ, f, n, p, β, γ, µ) a generic constant that we allow to vary each line. For ε > 0 fixed, we exploit the dominated convergence theorem to let δ 0 and get C u p 2 u i 2 T ε(u i ) φ 2 x x 0 µ x y γ (2.25) u p 2 u i T ε (u i ) x x 0 µ φ2 x y γ u p 2 u i T ε (u i ) φ 2 x x 0 µ x y γ u p 2 u i T ε (u i ) φ φ x x 0 µ x y γ f i (x) T ε (u i ) x x 0 µ x y γ φ2. We use Young s inequality ab ϑa 2 b2 and the fact that T 4ϑ ε(t) t β and get: u p 2 u i T ε (u i ) x x 0 µ φ2 x y γ u p 2 2 u i χ { ui ε} φ x y γ 2 x x 0 µ 2 u i β 2 u p 2 u i 2 χ { ui ε} ϑ x y γ x x 0 µ u i β φ2 C ( u u(x 0 ) 4ϑ x y γ2 x x 0 µ p β u p 2 u i 2 χ { ui ε} ϑ x y γ x x 0 µ u i β φ2 C 4ϑ, u p 2 2 u i 2 β 2 x y γ2 2 x x 0 µ 2 ) p β φ 2 φ

11 REGULARITY AND COMPARISON PRINCIPLES since we assumed that γ < n 2 and µ < min{α M, s n }(p ). (p )s Exploiting again Young s inequality we get: u p 2 u i T ε (u i ) x x 0 µ x y γ φ2 u p 2 2 u i χ { ui ε} φ x y γ 2 x x 0 µ 2 u i β 2 u p 2 u i 2 χ { ui ε} ϑ x y γ x x 0 µ u i β φ2 C 4ϑ ϑ C ϑ ϑ u u(x 0 ) p β φ 2 x y γ x x 0 µ2 u p 2 u i 2 χ { ui ε} x y γ x x 0 µ u i β φ2 x y γ x x 0 2 φ2 u p 2 u i 2 χ { ui ε} x y γ x x 0 µ u i β φ2 C 4ϑ. u p 2 2 u i 2 β 2 x y γ 2 x x 0 µ2 2 since we assumed that γ < n 2 and µ < min{α M, s n }(p ). We (p )s now use the fact that φ and get: ρ u p 2 u i T ε (u i ) φ φ x x 0 µ x y γ = 2 u p 2 2 u i χ { ui ε} u p 2 2 u i 2 β 2 φ ρ x x 0 µ 2 x y γ 2 u i β 2 x x 0 µ 2 x y γ 2 u p 2 u i 2 χ { ui ε} ϑ x x 0 µ x y γ u i β φ2 C ( u u(x0 ) ) p β ϑ x y γ x x 0 µ p β u p 2 u i 2 χ { ui ε} ϑ φ 2 C x y γ u i β ϑ, since we assumed that γ < n 2 and µ < min{α M, s n }(p ). (p )s Finally we have: f i (x) T ε (u i ) x x 0 µ x y γ φ2 ( ) ( C f m m ( ) m ) m C x x 0 µ x y γ φ

12 2 BERARDINO SCIUNZI by assump- since µ <, γ < n 2 and f W,m () for some m > tion. n 2 µ Taking into account (2.25), exploiting the above estimates and evaluating T ε, we get u p 2 u i 2 ( G u i β x x 0 µ x y γ ε (u i ) β G ε(u i ) Cϑχ { ui ε}) φ 2 C. u i Since for for ϑ small (G ε(u i ) β Gε(u i) u i Cϑχ { ui ε}) is strictly positive we conclude that u p 2 β u i 2 x x 0 µ x y γ φ2 and the thesis follows by the definition of φ. u p 2 u i 2 u i β x x 0 µ x y γ φ2 C, Corollary 2.2. Let u W,p () be a solution to (.). Assume that (.3) holds and let < p < 3. Assume that f W,m () for some m n 2 µ and consider a critical point x 0 Z u with. Then we have D 2 u 2 (2.26) x x 0 C, η B ρ (x 0 ) with C = C(p, n, f, u, x 0 ) and for any η such that η < η := n ( 2 min{α M, s n (p )s } 2). Since µ, the result holds in particular if f W,n (). As a consequence we also get that u W 2,2 loc (). Proof. The proof follows by directly by Theorem 2. and (.4) since we have that p 3 < 0 by assumption. 3. Local summability of the weight We prove in this section a summability property of u, extending the results of [5] to the case of vanishing source terms. We will assume that f satisfies the condition (I µ ) and prove the following: Theorem 3.. Let < p < and let u W,p () be a solution of (.). Assume that f satisfies (.3) and assume that (I µ ) holds. Then, for any x 0 Z u and for some ρ = ρ(x 0 ) > 0, we have (3.27) C. u (p ) x y (n 2) B ρ (x 0 )

13 REGULARITY AND COMPARISON PRINCIPLES 3 Namely B ρ (x 0 ) u t x y C, γ with 0 t < p, γ < n 2 if n 3 and γ = 0 if n = 2 and C = C(t, γ, p, f, ρ, x 0 ) not depending on y. Proof. Consider ϕ = u t ε x x 0 µ δ Gδ(f) H δ ( x y ) φ 2 f with ε, δ > 0, H δ and G δ defined according to (2.22), t < p and 0 < µ < µ with µ defined in (.5) such that (.2) and (.3) hold, by (I µ ). Here, as above, we assume that the ball is contained in, and we consider a cut-off function φ = φ ρ, in such a way that φ ρ Cc (), φ ρ = in B ρ (x 0 ) and φ ρ 2. In particular ρ we may and do assume that (I µ ) holds in. We will write φ instead of φ ρ. We use ϕ as test function in (2.6) and get (3.28) = 2 f(x)ϕ = u p 2 ( u, ϕ) = t u t u p 2 ( u, u ) Gδ(f) H ( u t ε) 2 x x 0 µ δ φ 2 δ f µ x x 0 µ u p 2 ( u, x x 0 ) Gδ(f) H ( x x 0 µ δ) 2 u t δ φ 2 ε f u p 2 ( u, x H δ ( x y )) u t ε u p 2 ( u, φ) u t ε ( Gδ (t) t x x 0 µ δ Gδ(f) f x x 0 µ δ Gδ(f) H δ φ f ) χ {δ f 2δ} t=f u t ε u p 2 ( u, f) H x x 0 µ δ φ 2, δ φ 2

14 4 BERARDINO SCIUNZI that gives C f ϕ N u p u t u i ( u t ε) H δ ( x y ) 2 x x 0 µ δ G δ(f) φ 2 f i= u p x x 0 µ u t ε G δ(f) H δ ( x y ) φ 2 f u p H δ ( x y ) u t ε u p φ u t ε u p f u t ε x x 0 µ δ G δ(f) φ 2 f x x 0 µ δ G δ(f) H δ ( x y )φ f x x 0 µ δ Hδ( x y ) φ 2, f where we have estimated the last term exploiting the fact that ( G δ(t) ) 4 t=f for δ f 2δ. t f Here and in the following we denote with C = C(x 0, ρ, f, n, p, t, γ, µ) a generic constant that we allow to vary each line. We now let δ 0 and, by the dominated convergence theorem and (.2), (.3) (see (I µ )), we get: (3.29) C u t ε x y γ φ2 f u t ε x x 0 µ N u p u t u i ( u t ε) 2 i= u p x x 0 µ u t ε x y γ φ2 x x 0 µ x y γ φ2 u p u t ε x x 0 µ φ2 x y γ u p φ u t ε u p f u t ε x x 0 µ x y φ γ x x 0 2µ x y γ φ2. x y γ φ2

15 REGULARITY AND COMPARISON PRINCIPLES 5 The reader can easy check that, since µ < and γ < n 2 and ε > 0, then the dominated convergence theorem applies. Let us now estimate the terms on the righthand side of (3.29). We have: C ϑ N u p u t u i ( u t ε) 2 i= x x 0 µ x y γ φ2 φ 2 ( u t ε) 2 x y γ 2 x x 0 u pt 2 D 2 u µ ( u t ε) 3 2 x y γ 2 ( u t ε) x y γ C ϑ ϑ ( u t ε) x y γ φ 2 x x 0 µ u (p 2) (2t p) D2 u 2 φ 2 x y γ x x 0 µ φ 2 x x 0 µ C ϑ, exploiting Theorem 2. (note that (.2) is enough) with β = 2 t p. Note that the assumption t < p implies β <. The assumption t < p also yields C C u p x x 0 µ u t ε x y γ φ2 x x 0 µ x y γ φ2 since µ < (see (.5)) and γ < n 2. Moreover u p u t ε x x 0 µ φ2 x y γ C C, x x 0 µ φ2 x y γ since µ < and γ < n 2. Also we have: u p φ u t ε C ρ C x x 0 µ φ φ x x 0 µ x y γ x x 0 µ φ x y γ C, x y γ φ

16 6 BERARDINO SCIUNZI since t < p, µ <, γ < n 2. Note that here the constant that we get depends on ρ as stated in the theorem. We also have u p f u t ε ( C f ) f m x x 0 2µ x y γ φ2 x x 0 2µ x y γ φ2 m ( ( ) m ) m C, x x 0 2µ x y γ exploiting that f W,m () for some m > n (see (.2)) and the 2( µ) fact that t < p and µ < and γ < n 2 (γ = 0 is n = 2). Collecting the above estimates, by (3.29), we get ( Cϑ) ( u t ε) x y (3.30) γ x x 0 µ φ2 C. We now take ϑ small, say ϑ small such that in (3.30) we have ( Cϑ) 2, and get ( u t ε) x y γ x x 0 µ φ2 C, that gives the thesis by letting ε 0 and exploiting Fatou s Lemma. Corollary 3.2. Let p 3 and let u W,p () be a solution of (.). Assume that (.3) and (I µ ) hold and assume that f W,m () for some m > n 2 µ. Then, for x 0 Z u (and for some ρ = ρ(x 0 ) > 0) and for any q < p p 2, we have: D 2 u q (3.3) x x 0 τ C, for any B ρ (x 0 ) (3.32) τ < τ := n 2 q 2 µ, with µ defined in (.5). The result holds in particular if f W,n (). Proof. We can rewrite any τ < τ as: τ := γ q 2 (µ ε)

17 REGULARITY AND COMPARISON PRINCIPLES 7 for some γ < n 2 if n 3, γ = 0 if n = 2 and ε > 0 (small). Then, for any 0 β < and < q < 2, we have D 2 u q x x 0 τ dx B ρ(x 0 ) ( C, B ρ(x 0 ) B ρ(x 0 ) u (p 2 β) q 2 D 2 u q x x 0 q 2 (γµ ε) u (p 2 β) D 2 u 2 x x 0 γµ ε dx u (p 2 β) q 2 x x 0 γ 2 q 2 ) q 2 ( Bρ(x0) dx ) 2 q u (p 2 β) q 2 q x x0 γ dx 2 where we used Holder inequality with exponents 2 and 2, Theorem q 2 q 2. with y = x 0 (since we assumed that f W,m () for some m > n 2 µ ) and Theorem 3.. Note that, to apply Theorem 3., we can choose 0 < β < such that (p 2 β) q 2 q assumption q < p p 2. < p because of the 4. The strong comparison principle The summability properties of u obtained in Theorem 3. allows to prove a weighted Sobolev inequality that we recall here below: Theorem 4. ([5]). Let p > 2 and u C () be a solution of (.). Assume that f satisfies (.3) and (I µ ) (see (.2) and (.3)). Setting ρ = u p 2, for any, we have (4.33) with t = p p 2 and y. r, p 2 p ρ t x y C, γ < r <, γ < n 2 if n 3 while γ = 0 if n = 2, Assuming in the case n 3 with no loose of generality that γ > n 2t, then it follows that the space H,2 0 (, ρ) is continuously embedded in L q, for q < 2 (p), where 2 (p) = 2 n p 2 p n. More precisely there exists a constant C S such that (4.34) w L q ( ) C S w L 2 (,ρ) = C S ( w 2 ρ for any w H,2 0 (, ρ) and q < 2 (p). Moreover the embedding is compact. ) 2,

18 8 BERARDINO SCIUNZI Proof. Once Theorem 3. is proved then (4.33) follows by a simple covering argument. We can therefore exploit (4.33) to repeat verbatim the proof of Theorem 3. in [5]. Finally the embedding is compact by [2, Lemma 5.]. As a consequence we have Theorem 4.2. Let u, v C () such that p u f(x) p v f(x) in. Assume that u or v is a solution to (.) with 2n2 n2 < p <, and assume that f satisfies (.3) and (I µ ) (see (.2) and (.3)). Then, for x 0 Z u fixed, if we assume that u v in B 5ρ (x 0 ), it follows that (4.35) sup (v u) C H inf (v u) B ρ (x 0 ) for some constant C H = C H (u, v, f, n, p, ρ, x 0 ). Proof. The proof follows by the Moser-type iteration scheme, as developed in [30]. The technique in [30] actually is mainly based on Sobolev embedding and works in our case thanks to Theorem 4.. The details can be found in [6, Theorem 3.3]. We are now ready to prove Theorem.3 and Theorem.4. Proof of Theorem.3: The proof is a consequence of the Harnack inequality in Theorem 4.2. Let us set C u,v := { x : u(x) = v(x) }. Since u and v are continuous, it follows that C u,v is closed. Let us consider now x 0 C u,v and ρ = ρ(x 0 ) > 0 such that B 5ρ (x 0 ). Note that actually u(x 0 ) = v(x 0 ) since u touches v from below at x 0. If u(x 0 ) 0 then, eventually reducing ρ, we have that u = v in B ρ (x 0 ) by the classical strong comparison principle [3] (see also [0]). If else u(x 0 ) = 0 = v(x 0 ), eventually reducing ρ, we exploit Theorem 4.2 to prove that u = v in B ρ (x 0 ). Therefore C u,v is also open and the thesis follows recalling that is connected. Proof of Theorem.4: Let us consider the case when u is a nonnegative solution to the equation (.0) and set f(x) := x σ g(u(x)). Since p u 0, by the strong maximum principle [32] it follows that u is strictly positive, since we assumed that it is not trivial. Therefore,

19 REGULARITY AND COMPARISON PRINCIPLES 9 g(u(0)) > 0 since g(s) > 0 for s > 0 and, by the assumption 0 σ < µ, it follows that f satisfies the condition (I µ ) (it is easy to check the validity of (.2)). Therefore Theorem 3. holds and Theorem 4. follows as well. This allows to repeat verbatim the proof of [6, Theorem 3.3] and deduce also in this case the validity of the Harnack inequality, namely Theorem 4.2. The proof now can be finished arguing exactly as in the proof of Theorem Examples Example 5.. Let < p < and consider u C,α (B) weak solution to p u = u q in B (5.36) u > 0 in B u = 0 on B, where B is the unit ball in R n centered at zero. Then, by [4, 5], it follows that u is radial and radially decreasing with Z u := { u = 0} {0}. Therefore, arguing as in [26], by l Hopital s rule we have u x p and D 2 u x 2 p p. Example 5.2. Let p >. Then the function solves and u(x,..., x N ) = x p p u =, Z u = {x = 0}. Remark 5.3. It is easy to see that, for p 3, the solution given in Example 5.2 is no more regular than what we obtained in Corollary (3.2). On the contrary, the solutions in Example 5. are more regular (just use polar coordinates to see this). We might guess that the geometry of the critical set Z u plays a role. Proposition 5.4. Let u C,α () be a weak solution of (.) in R n. Suppose that x 0 Z u, namely u(x 0 ) = 0, and suppose that: for some ρ, ϑ > 0 we have that f ϑ in B ρ (x 0 ). Then (in a neighborhood of x 0 ) we have α p. In particular, for p > 2, we have that u / C 2 (). p

20 20 BERARDINO SCIUNZI Proof. Let us consider the balls B ρ (x 0 ) and centered in x 0 with 2ρ ρ. Consider a cut-off function φ ρ such that, φ ρ 0 in, φ ρ Cc () and { φρ in B ρ (5.37) n ρ (x 0 ) φ ρ c in B ρ n 2ρ (x 0 ) \ B ρ (x 0 ). Using φ ρ as test-function in (.) we get (5.38) u p 2 ( u, φ ρ ) dx = We have that: u p 2 ( u, φ ρ ) dx so that (5.39) On the other hand f φ ρ dx. u u(x 0 ) p φ ρ dx c ρα(p ) ρ n ρ n c ρ α(p ) u p 2 ( u, φ ρ ) dx ρ 0 0 if (α(p ) ) > 0. (5.40) f φ ρ dx ϑ φ ρ dx ϑ B > 0 for ρ sufficiently small. Therefore, for (α(p ) ) > 0, we get a contradiction by (5.38), (5.39) and (5.40). As a consequence we get that near x 0 necessarily (α(p ) ) 0 and the thesis. Note now that, if p > 2, then < so that u / p C2 (). References [] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations. Ann. of Math., 30 (), 989, pp [2] D. Castorina, P. Esposito and B. Sciunzi, Degenerate elliptic equations with singular nonlinearities. Calc. Var. and PDE, 34 (3), 2009, pp [3] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5(4), 998, pp [4] L. Damascelli, F. Pacella, Monotonicity and symmetry of solutions of p- Laplace equations, < p < 2, via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26(4), 998, pp [5] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-laplace equations. J. Differential Equations, 206(2), 2004, pp [6] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations. Calc. Var. Partial Differential Equations, 25 (2), 2006, pp

21 REGULARITY AND COMPARISON PRINCIPLES 2 [7] E. De Giorgi, Sulla differenziabilità e l analicità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino, serie 3a, 3, 957, pp [8] E. Di Benedetto, C α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 983, pp [9] F. Duzar, G. Mingione Gradient estimates via nonlinear potentials. Am. J. Math., 33, 20, pp [0] D. Gilbarg, N. and Trudinger, S, Elliptic partial differential equations of second order. Reprint of the 998 Edition, Springer. [] T. Kuusi, G. Mingione, Universal potential estimates. J. Funct. Anal., 262, 202, pp [2] T. Ivaniec and J. Manfredi, Regularity of p-harmonic functions on the plane. Revista Matematica Iberoamericana, 5(-2), 989, pp. 9. [3] O. A. Ladyzhenskaya, N. N. Ural tseva, Linear and quasilinear elliptic equations. Academic Press, 968. [4] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 2(), 988, pp [5] H. Lou, On singular sets of local solutions to p-laplace equations, Chinese Annals of Mathematics, Series B29. 5, 2008, pp [6] N.G. Meyers, G. and J. Serrin, H = W. Proc. Nat. Acad. Sci. U.S.A., 5, 964, pp Papers in memory of Ennio De Giorgi. [7] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math., 5 (4), 2006, pp [8] G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (5)., 6(2), 2007, pp [9] G. Mingione, Gradient estimates below the duality exponent. Math. Ann., 346, 200, pp [20] G. Mingione, Gradient potential estimates. J. Eur. Math. Soc., 3, 20, pp [2] J.K. Moser, On Harnack s theorem for elliptic differential equations. Comm. on Pure and Applied Math., 4, 96, pp [22] J. Nash, Continuity of solutions of elliptic and parabolic equations. Amer. J. Math., (80) 958, pp [23] P. Pucci, J. Serrin, The maximum principle. Birkhauser, Boston (2007). [24] P. Pucci and R. Servadei, Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. Indiana Univ. Math. J., 57(7), 2008, pp [25] B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations. NoDEA Nonlinear Differential Equations Appl., 4(3-4) (2007), [26] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J., 49(3), 2000, pp [27] E. V. Teixeira, Regularity for quasilinear equations on degenerate singular sets. Math. Ann. DOI: 0.007/s [28] E. V. Teixeira, Sharp regularity for general Poisson equations with borderline sources. J. Math. Pures Appl., (9). 99(2), 203, pp [29] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations. 5(), 984, pp [30] N.S. Trudinger, Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa (3), 27, 973, pp [3] N.S. Trudinger, On the regularity of generalized solutions of linear nonuniformly elliptic equations, Arch. Rational Mech. Anal., 42, 97, pp

22 22 BERARDINO SCIUNZI [32] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 2(3), 984, pp Dipartimento di Matematica, UNICAL,, Ponte Pietro Bucci 3B,, Arcavacata di Rende, Cosenza, Italy address: sciunzi@mat.unical.it

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