Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 2, 69-81 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.31225 Harnack Inequality and Continuity of Solutions for Quasilinear Elliptic Equations in Sobolev Spaces with Variable Exponent Azeddine Baalal and Abdelbaset Qabil Department of Mathematics - Laboratory MACS Faculty of Sciences Aïn Chock, University of HASSAN II B.P. 5366, Casablanca - Morocco Copyright c 2014 Azeddine Baalal and Abdelbaset Qabil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We investigate regularity properties of solutions to the quasilinear elliptic equations in Sobolev spaces with variable exponent, we prove the Harnack s inequality and continuity of solutions. Mathematics Subject Classification: 35J62, 49N60, 35J60, 35J25. Keywords: Quasilinear elliptic equation, variable exponent, Caccioppoli estimate, Moser s iteration method, Harnack inequality 1 Introduction In the present paper we study the regularity of boundary points for solutions to the quasilinear elliptic equations: div Ax, u+bx, u =0, 1 Our purpose is to establish the Harnack s inequality ess sup u Cess inf u + R 2 Bx,R Bx,R
70 A. Baalal and A. Qabil where C is independent of u and the ball B = Bx 0,R for x 0, R>0 and we prove the continuity of solutions for 1. Harnack s inequality and other regularity results for 1 require additional assumptions on the function p.; see the counterexamples in [6]. The so called logarithmic Hölder continuity condition seems to be the right one for our purposes. This condition was originally introduced by Zhikov [15] in the context of the Lavrentiev phenomenon for solutions of 1, and it has turned out to be a useful tool in regularity and other applications, see, e.g., [1, 2]. For the existence and uniqueness of solutions u W 1,px where 1 <px < d for all x, of the variational Dirichlet problem associated with the quasilinear elliptic equation 1 see [4], these solutions are obtained by the p.- obstacle problem. A typical example for the operator A and B are Ax, u = u px 2 u and Bx, u = u px 2 u respectively, for all x R d thus Δ px u = div u px 2 u. Our problem has been studied in many paper see e.g. [11, 12]. Olli Toivanen [11] proved that this problem has a solution when the operator B depends on x, u, u where δx = px 1. The main aim of this section is to generalize the condition on δx, We are interested in the case that the operator B depends only on x and u, satisfying the previous hypothesis H3, where px 1 δx <p x, knowing that the study of the case where δ satisfies the condition 0 δx <px 1 is already investigated in several articles. The contribution of this paper is to verify the Harnack principale for a weak solutions of quasilinear elliptic equations 1 by using the trick of modified test functions under our assumptions H3 below. In the first section, we introduce some generalization and position of the problem. In second section we give some basic facts about variable exponent spaces and a rough overview of properties of solutions of the prototype equality. In section 3, we generalize, with detailed proofs, Harnack s inequality 2 to all quasilinear elliptic equations 1 with growth conditions of a non-standard form. In last section, we present the concluding remarks. 2 Some preliminaries We start this section with some definitions and main results of Lebesgue spaces with variable exponent, and Sobolev spaces modeled upon them. For each open bounded subset of R d d 2, we define the Lebesgue space with variable exponent L p. as the set of all measurable functions p :
Harnack inequality in W 1,p. 71 ]1, + [ called a variable exponent and we denote p := ess inf x px and p + := ess sup x px. We introduce also the convex modular ϱ px u = u px dx. If the exponent is bounded, i.e., if p + <, then the expression u p. = inf{λ >0:ϱ p. u λ 1} defines a norm in L p., called the Luxemburg norm. One central property of L p. is that the norm and the modular topologies coincide,i.e., ϱ p. u n 0 if and only if u n p. 0. We denote by L p. the conjugate space of L p. where 1 + 1 =1. px p x Proposition 1 Generalized Hölder inequality [14] and v L p., we have uvdx 1 p + 1 p u p. v p.. We define the variable exponent Sobolev space see [9], [5],[8], [14] by with the norm W 1,p. = {u L p. : u L p. }. For any u L p. u 1,p. = u p. + u p. u W 1,p.. The local Sobolev space W 1,p. loc consists of functions u that belong to W 1,p. loc U for all open sets U compactly contained in. The Sobolev space with zero boundary values, W 1,p. 0, is defined as the completion of C0 in the norm of W 1,p.. Let p x bethe Sobolev conjugate exponent of px defined by p x = { dpx d px for px <d, + for px d. We assume further on that, there exist positive constant C such that the function p satisfies logarithmic Hölder continuity condition if : { C>0: px py C for x y < 1 log x y 2, 1 <p p + <d.
72 A. Baalal and A. Qabil Proposition 2 The p.-poincar?e inequality Let be a bounded open set and let p : [1, [ satisfy There exists a constant C, depending only on p. and, such that the inequality u p. C u p. u W 1,p. 0. Lemma 2.1 Sobolev inequality [7] Let be a bounded open set and u in W 1,p. 0. There exists a constant C such that u dpx dx dpx C u px dx 1 px 3 Proposition 3 Assuming p > 1, the spaces W 1,p. and W 1,p. 0 are separable and reflexive Banach spaces. Throughout the paper we suppose that the functions A : R d R d R d is a Carathéodory function satisfying the following assumptions: H1 Ax, ξ β[kx+ ξ px 1 ]; H2 Ax, ξξ ν ξ px ; for a.e. x, all ξ R d, where kx is a positive bounded function lying in L p x and β,ν > 0. In this paper we suppose that the function B : R d R R is given Carathéodory function and the following condition is satisfied: H3 Bx,ζ gx+ ζ δx ; for a.e. x, all ζ R d, where g is a positive bounded function lying in L p x and px 1 δx <p x. Remark 2.1 Under the assumption Harnack s inequality and local Hölder continuity follow from Moser or De Giorgi-type procedure; see [10, 2, 3]. An interesting feature of this theory is that estimates are intrinsic in the sense that they depend on the solution itself. For example, supersolutions are assumed to be locally bounded and Harnack-type estimates in [2] depend on this bound. Definition 2.1 We say that a u W 1,p. loc is a p.-solution of 1 in provided that for all ϕ W 1,p. 0 if, Ax, u ϕdx + Bx, uϕdx =0.
Harnack inequality in W 1,p. 73 Definition 2.2 A function u W 1,p. loc is termed p.-supersolutions of 1, if and only if, for all non-negative functions ϕ W 1,p. 0 we have, Ax, u ϕdx + Bx, uϕdx 0. A function u is a p.-subsolution in if u is a p.-supersolution in, and a solution in if it is both a super- and a p.-subsolution in. 3 Harnack inequality and continuity of solutions to quasilinear elliptic equations The Harnack inequality is a very important estimate in the study of p.- solutions of quasilinear elliptic equations. 3.1 Main result We start by adapting a standard Caccioppoli type estimate for p.-supersolution of 1. Then we use the Caccioppoli estimate to show that for a fixed, nonnegative p.-supersolution u, the inequality 2. The Harnack inequality is indispensable as a tool in the qualitative theory of second-order elliptic equations. In particular, it implies continuity of weak solutions see [11, 13]. By non-linearity we mean that if p 2 then the weak solutions do not form a linear space. However the set of weak solutions is closed under constant multiplication. By celebrated De Giorgi s method and Moser s iteration the weak solutions are locally Hölder continuous and satisfy Harnack s inequality. Remark 3.1 Our notation is rather standard. Various constants are denoted by C and the value of the constant may differ even on the same line. The quantities on which the constants depend are given in the statements of the theorems and lemmas. Lemma 3.1 Let E be a measurable subset of R d. For all nonnegative measurable functions ψ and ϕ defined on E, ψϕ p E dx ψdx + ψϕ px dx E E E
74 A. Baalal and A. Qabil The following Caccioppoli estimate is the key result of this paper; and it is a modification of [[7], Lemma 3.2]. The new feature in the estimate is the choice of a test function which includes the variable exponent. Lemma 3.2 Caccioppoli estimate Suppose that u is a nonnegative p.-supersolution in. Let E be a measurable subset of and η C0 such that 0 η 1. Then for every γ 0 < 0 there is a constant C depending on p and γ 0 such that the inequality: η p+ u γ 1 u p dx C u γ+px 1 η px + η p + u γ 1 4 E +u γ+px 1 dx + η p+ B 1 4R u γ+δx dx holds for every γ<γ 0 < 0 and px 1 δx <p x. Proof 1 Let θ = p +. We want to test with the function ϕ = η θ u γ. To this end we show that ϕ W 1,p. 0. Since η has a compact support in, it is enough to show that ϕ W 1,p.. We observe that ϕ L px since u γ η θ R γ. Furthermore, we have ϕ = γη θ u γ 1 u + θη θ 1 u γ η Using the fact that u is a p.-supersolution and ϕ is a nonnegative test function we find that 0 Ax, u ϕdx + Bx, uϕdx B 4R = γη θ u γ 1 Ax, u udx + θη θ 1 u γ Ax, u ηdx + Bx, uη θ u γ dx We denote the left-hand side of the next inequality by I. Since γ is a negative number this implies by the structural conditions H1, H2 and H3 that Z I = γ 0 ν η θ u γ 1 u px dx px 1 δx β θη ZB4R θ 1 u kx+ u γ η dx + gx+u η ZB4R θ u γ dx =I 1 =I z Z } 2 =I z Z } Zz } 3 βθ η θ 1 u γ u px 1 η dx +βθ η θ 1 u γ kx η dx + gx+u δx η θ u γ dx Using Young s inequality, 0 <ε 1, we obtain the first estimate I 1 1 px p ε px 1 η θ θ p 1 x u γ+px 1 px η + ε η θ p x u γ γ+px 1 px u xdx px 1 1 ε B θ 1 η θ px u γ+px 1 η px dx + ε η θ u γ 1 u px dx 4R 1 ε B θ 1 u γ+px 1 η px dx + ε η θ u γ 1 u px dx 4R
Harnack inequality in W 1,p. 75 Next we estimate the last tow integrals I 2 and I 3. To estimate the integral I 2, we denote 0 v = η + η and k is a positive bounded function there exist a constant M>0, and by Young s inequality we have I 2 = η θ 1 u γ kx η dx M vη θ 1 u γ dx M 1ε θ 1 η θ px u γ+px 1 v px dx + ε η θ u γ 1 dx M 1ε θ 1 u γ+px 1 v px dx + ε η θ u γ 1 dx M 1ε θ 1 u γ+px 1 1 + η px dx + ε η θ u γ 1 dx Using the inequality ϕ + ψ p. 2 p. 1 ϕ p. + ψ p., for p. 1, I 2 1 ε B θ 1 u γ+px 1 2 px 1 1 + η px dx + ε η θ u γ 1 dx 4R B 4R 2θ 1 u γ+px 1 1 + η px dx + ε η θ u γ 1 dx ε θ 1 2 ε θ 1 u γ+px 1 η px dx + 2 ε θ 1 u γ+px 1 + ε η θ u γ 1 dx To estimate the integral I 3, and by the assumption g is a positive bounded function there exist a N>0such that I 3 = gx+u δx η θ u γ dx N + u δx η θ u γ dx =I 3 { }}{ η θ 1 u γ+δx dx + N η θ 1 u γ dx By Young s inequality we have I 3 1 ε B θ 1 u γ+px 1 dx + ε η θ u γ 1 dx 4R Thus I 3 η θ 1 u γ+δx dx + 1 ε θ 1 u γ+px 1 dx + ε η θ u γ 1 dx
76 A. Baalal and A. Qabil Therefore γ 0 ν θβε η θ u γ 1 u px dx θβ 2θ 1 +1 u γ+px 1 η px dx ε θ 1 B 4R +εθβ +1 η θ u γ 1 dx + 2θ 1 θβ +1 u γ+px 1 dx + η θ 1 u γ+δx dx ε θ 1 By choosing ε = min{1, γ 0 ν 2θβ } η θ u γ 1 u px dx C 1 u γ+px 1 η px dx + C 2 + C 4 η θ 1 u γ+δx dx η θ u γ 1 dx + C 3 u γ+px 1 dx + Where C 1 =2 θ 1 +1 2θβ γ 0 ν, C2 =1+ 1, C θβ 3 =2 θ 1 + 1 2θβ θ, θβ γ 0 ν and C 4 = 2. γ 0 ν We take C = C i for i =1, 2, 3, 4. and we have η p + u γ 1 u px dx C u γ+px 1 η px + η p + u γ 1 +u γ+px 1 dx + η p+ 1 u γ+δx dx By the lemma 3.1 we obtain η θ u γ 1 u p dx η θ u γ 1 dx + η θ u γ 1 u px dx and using the previous inequality we obtain the claim. η p+ u γ 1 u p dx C u γ+px 1 η px + η p + u γ 1 E +u γ+px 1 dx + η p+ B 1 4R u γ+δx dx So the proof of lemma is achieved. 3.2 Weak Harnack Inequality In this section we prove a weak Harnack inequality for p.-supersolutions to 1.
Harnack inequality in W 1,p. 77 Throughout this subsection we write v = u + R where u is a nonnegative p.-supersolution and 0 R 1. We start by the following technical lemma that is need later. These results are mainly from [7] Lemma 3.3 If the exponent p. is log-hölder continuous, then R px where x E BR and R>0. CR p E Lemma 3.4 Let f be a positive measurable function and assume that the exponent p. is log-hölder continuous. Then + B p 4R dx C f p+ B p 4R for any s>p + p f p L s Now we have everything ready for the iteration. We write Φf,q, = f q dx 1 q for a nonnegative measurable function f. The point is that the Moser iteration technique used in [7] remains valid under our consideration. Lemma 3.5 Let u is a nonnegative p.-supersolution of 1 in and let R ρ<r 3R. Then the inequality Φv,qτ, C 1 p + τ 1 + τ τ p r r ρ + τ d Φv,τ d 1,B ρ 5 holds for every τ<0and 1 <q< d. The constant C depends on d,p, and the L q s -norm of u with s>p + p and all structure constants and functions of H1,H2 and H3 hypothesis. Proof 2 Let θ + = p + and θ = p, we take γ = τ θ + 1. In 4 of the lemma 3.2 we have + v τ θ u θ dx 6 η θ =I 1 =I 2 {}}{{}}{ C η θ + u τ θ + u τ θ +px η px + u τ θ +px + η θ+ 1 u τ θ +1+δx dx Now we take the test function η C 0 with 0 η 1, η =1inB ρ, and η Cr Rr ρ
78 A. Baalal and A. Qabil Next we went to estimate the integral I 2 by the integral v qτ dx 1 q 7 Using lemma 3.3,3.4, the first integral I 1 is estimated by 7 see [7]. Finally, for the second integral I 2 we have by Hölder s inequality u τ θ +px dx v τ θ +px dx C v q px θ dx C 1+ v q θ + θ L q s 1 q Br 1 q Br 1 v qτ q dx 1 v qτ q dx On the other hand and since px 1 δx <p x η θ+ 1 u τ θ +1+δx dx v τ θ +1+δx dx 1 C v q δx θ +1 q dx v qτ dx Br C 1+ v q θ + θ L q s 1 q 1 v qτ q dx Br Therefore, the second integral I 2 is estimated by 3. In lemma 2.1 we take u = v τ θ η θ θ and we use the inequality: We obtain v dτ B ρ a + b p. 2 p. 1 a p. + b p., p. 1, d CR θ τ C v θ η θ dθ d θ dx v τ θ η θ θ θ dx B r C τ θ v τ θ u θ dx + C η θ+ Using inequality 4 we have 1 q v τ θ+ θ η η θ dx
Harnack inequality in W 1,p. 79 v dτ d B ρ τ C v θ η θ dθ d θ dx CR θ v τ θ η θ θ θ dx B r C τ η θ θ+ u τ θ + u τ θ +px η px + v τ η θ+ θ η θ dx B r + C u τ θ +px + η θ+ 1 u τ θ +1+δx dx C1 + τ θ+ 1+ v q θ + θ L q s Finally, since τ<0we have Φv,qτ, C 1 p + τ 1 + τ So the proof of lemma is achieved. 1 q τ r r ρ p r r ρ + τ θ + v qτ dx 1 q d Φv,τ d 1,B ρ The proofs of the following results can be found in [11]. and [7], respectively Lemma 3.6 Assume that u is a nonnegative p.-supersolution of 1 in and s>p + p. Then there exist constants q 0 > 0 and C depending on d, p, and L q s -norm of u such that: Φv,q 0,B 3R CΦv, q 0,B 3R 8 Theorem 3.7 Weak Harnack inequality Let u be a non-negative p.- supersolution of 1 in and 1 <q< d. Then u q 0 dx 1 q 0 C ess inf ux+r 9 B x B R 2R Where q 0 is the exponent from Lemma 3.5 and C depends on d,p,q and the L q s -norm of u with s>p + p and all structure constants and functions of H1,H2 and H3 hypothesis. Theorem 3.8 Let u be a non-negative p.-subsolution of 1 in and 1 <q< d. Then ess sup ux C u t dx 1 t + R 10 x B R B 2R Where t>0 and C depends on d,p,q and the L q s -norm of u with s> p + p.
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