C O M M U N I C AT I O N S I N M AT H E M AT I C S

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1 VOLUME 23/2015 No. 1 ISSN (Print ISSN (Online C O M M U N I C AT I O N S I N M AT H E M AT I C S Editor-in-Chief Olga Rossi, The University of Ostrava & La Trobe University, Melbourne Division Editors Ilka Agricola, Philipps-Universität Marburg Attila Bérczes, University of Debrecen Anthony Bloch, University of Michigan George Bluman, The University of British Columbia, Vancouver Karl Dilcher, Dalhousie University, Halifax Stephen Glasby, University of Western Australia Yong-Xin Guo, Eastern Liaoning University, Dandong Haizhong Li, Tsinghua University, Beijing Vilém Novák, The University of Ostrava Geoff Prince, La Trobe University, Melbourne Gennadi Sardanashvily, M. V. Lomonosov Moscow State University Thomas Vetterlein, Johannes Kepler University Linz Technical Editors Lukáš Novotný, The University of Ostrava Jan Šustek, The University of Ostrava Available online at

2 Communications in Mathematics 23 ( Copyright c 2015 The University of Ostrava 33 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations Albo Carlos Cavalheiro Abstract. In this paper we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations (v(x u q 2 [ u D j ω(xaj(x, u, u ] = f 0(x D jf j(x, in in the setting of the weighted Sobolev spaces. 1 Introduction In this paper we prove the existence and uniqueness of (weak solutions in the weighted Sobolev space X = W 2,q (, v W 1,p 0 (, ω (see Definition 4 for the Navier problem Lu(x = f 0 (x n u(x = 0, u(x = 0, where L is the partial differential operator Lu(x = (v(x u q 2 u D jf j (x, in on on [ D j ω(xaj (x, u(x, u(x ] where D j = / x j, is a bounded open set in R n, ω and v are two weight functions, is the usual Laplacian operator, 1 < p, q < and the functions A j : R R n R (j = 1,..., n satisfy the following conditions: 2010 MSC: 35J70, 35J60 Key words: degenerate nolinear elliptic equations, weighted Sobolev spaces, Navier problem (1

3 34 Albo Carlos Cavalheiro (H1 x A j (x, η, ξ is measurable on for all (η, ξ R R n (η, ξ A j (x, η, ξ is continuous on R R n for almost all x. (H2 There exists a constant θ 1 > 0 such that [A(x, η, ξ A(x, η, ξ ] (ξ ξ θ 1 ξ ξ p whenever ξ, ξ R n, ξ ξ, where A(x, η, ξ = (A 1 (x, η, ξ,..., A n (x, η, ξ (where a dot denote here the Euclidean scalar product in R n. (H3 A(x, η, ξ ξ λ 1 ξ p, where λ 1 is a positive constant. (H4 A(x, η, ξ K 1 (x + h 1 (x η p/p + h 2 (x ξ p/p, where K 1, h 1 and h 2 are non-negative functions, with h 1 and h 2 L (, and K 1 L p (, ω (with 1/p + 1/p = 1. By a weight, we shall mean a locally integrable function ω on R n such that ω(x > 0 for a.e. x R n. Every weight ω gives rise to a measure on the measurable subsets on R n through integration. This measure will be denoted by µ. Thus, µ(e = E ω(x dx for measurable sets E Rn. In general, the Sobolev spaces W k,p ( without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [1], [2] and [4]. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is disturbed in the sense that some degeneration or singularity appears. This bad behaviour can be caused by the coefficients of the corresponding differential operator as well as by the solution itself. The so-called p-laplacian is a prototype of such an operator and its character can be interpreted as a degeneration or as a singularity of the classical (linear Laplace operator (with p = 2. There are several very concrete problems from practice which lead to such differential equations, e.g. from glaceology, non-newtonian fluid mechanics, flows through porous media, differential geometry, celestial mechanics, climatology, petroleum extraction, reaction-diffusion problems, etc. A class of weights, which is particulary well understood, is the class of A p - weights (or Muckenhoupt class that was introduced by B. Muckenhoupt (see [11]. These classes have found many useful applications in harmonic analysis (see [13]. Another reason for studying A p -weights is the fact that powers of distance to submanifolds of R n often belong to A p (see [10]. There are, in fact, many interesting examples of weights (see [9] for p-admissible weights. In the non-degenerate case (i.e. with v(x 1, for all f L p (, the Poisson equation associated with the Dirichlet problem { u = f(x, in u(x = 0, on

4 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 35 is uniquely solvable in W 2,p ( W 1,p 0 ( (see [8], and the nonlinear Dirichlet problem { p u = f(x, in u(x = 0, on is uniquely solvable in W 1,p 0 ( (see [3], where p u = div( u p 2 u is the p- Laplacian operator. In the degenerate case, the weighted p-biharmonic operator has been studied by many authors (see [12] and the references therein, and the degenerated p-laplacian was studied in [4]. The following theorem will be proved in section 3. Theorem 1. Assume (H1 (H4. If (i v A q, ω A p (with 1 < p, q <, (ii f j /ω L p (, ω (j = 0, 1,..., n, then the problem (1 has a unique solution u X = W 2,q (, v W 1,p 0 (, ω. 2 Definitions and basic results Let ω be a locally integrable nonnegative function in R n and assume that 0 < ω(x < almost everywhere. We say that ω belongs to the Muckenhoupt class A p, 1 < p <, or that ω is an A p -weight, if there is a positive constant C = C p,ω such that ( ( p ω(x dx ω 1/(1 p (x dx C B B B B for all balls B R n, where denotes the n-dimensional Lebesgue measure in R n. If 1 < q p, then A q A p (see [7], [9] or [13] for more information about A p -weights. The weight ω satisfies the doubling condition if there exists a positive constant C such that µ(b(x; 2 r Cµ(B(x; r, for every ball B = B(x; r R n, where µ(b = B ω(x dx. If ω A p, then µ is doubling (see Corollary 15.7 in [9]. As an example of A p -weight, the function ω(x = x α, x R n, is in A p if and only if n < α < n(p 1 (see Corollary 4.4, Chapter IX in [13]. If ω A p, then ( p E C µ(e B µ(b whenever B is a ball in R n and E is a measurable subset of B (see 15.5 strong doubling property in [9]. Therefore, if µ(e = 0 then E = 0. Definition 1. Let ω be a weight, and let R n be open. For 0 < p < we define L p (, ω as the set of measurable functions f on such that f L p (,ω = ( f(x p ω(x dx 1/p <.

5 36 Albo Carlos Cavalheiro If ω A p, 1 < p <, then ω 1/(p 1 is locally integrable and we have L p (, ω L 1 loc ( for every open set (see Remark in [14]. It thus makes sense to talk about weak derivatives of functions in L p (, ω. Definition 2. Let R n be open, 1 < p < and ω A p. We define the weighted Sobolev space W k,p (, ω as the set of functions u L p (, ω with weak derivatives D α u L p (, ω, 1 α k. The norm of u in W k,p (, ω is defined by ( u W k,p (,ω = u(x p ω(x dx + 1/p D α u(x p ω(x dx. (2 1 α k We also define W k,p 0 (, ω as the closure of C 0 ( with respect to the norm W k,p (,ω. If ω A p, then W k,p (, ω is the closure of C ( with respect to the norm (2.1 (see Theorem in [14]. The spaces W k,p (, ω and W k,p 0 (, ω are Banach spaces and the spaces W k,2 (, ω and W k,2 0 (, ω are Hilbert spaces. It is evident that a weight function ω which satisfies 0 < c 1 ω(x c 2 for x (where c 1 and c 2 are constants, gives nothing new (the space W k,p 0 (, ω is then identical with the classical Sobolev space W k,p 0 (. Consequently, we shall be interested above in all such weight functions ω which either vanish somewhere in or increase to infinity (or both. In this paper we use the following results. Theorem 2. Let ω A p, 1 < p <, and let be a bounded open set in R n. If u m u in L p (, ω then there exist a subsequence {u mk } and a function Φ L p (, ω such that (i u mk (x u(x, m k, µ-a.e. on ; (ii u mk (x Φ(x, µ-a.e. on ; (where µ(e = ω(x dx. E Proof. The proof of this theorem follows the lines of Theorem in [6]. Theorem 3. (The weighted Sobolev inequality Let be an open bounded set in R n and ω A p (1 < p <. There exist constants C and δ positive such that for all u C 0 ( and all k satisfying 1 k n/(n 1 + δ, u L kp (,ω C u L p (,ω. Proof. See Theorem 1.3 in [5]. Lemma 1. Let 1 < p <. (a There exists a constant α p such that x p 2 x y p 2 y α p x y ( x + y p 2 y, for all x, y R n ;

6 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 37 (b There exist two positive constants β p, γ p such that for every x, y R n β p ( x + y p 2 x y 2 ( x p 2 x y p 2 y (x y γ p ( x + y p 2 x y 2. Proof. See [3], Proposition 17.2 and Proposition Definition 3. Let R n be a bounded open set and v A q, ω A p, 1 < p, q <. We denote by X = W 2,q (, v W 1,p 0 (, ω with the norm u X = u L p (,ω + u L q ( v. Definition 4. We say that an element u X = W 2,q (, v W 1,p 0 (, ω is a (weak solution of problem (1 if for all ϕ X we have u q 2 u ϕ v dx + ω A j (x, u(x, u(xd j ϕ dx = f 0 ϕ dx + f j D j ϕ dx. 3 Proof of Theorem 1 The basic idea is to reduce the problem (1 to an operator equation Au = T and apply the theorem below. Theorem 4. Let A : X X be a monotone, coercive and hemicontinuous operator on the real, separable, reflexive Banach space X. Then for each T X the equation Au = T has a solution u X. Proof. See Theorem 26.A in [16]. To prove the existence of solutions, we define B, B 1, B 2 : X X R and T : X R by B(u, ϕ = B 1 (u, ϕ + B 2 (u, ϕ, B 1 (u, ϕ = ωa j (x, u, ud j ϕ dx = B 2 (u, ϕ = T (ϕ = u q 2 u ϕv dx, f 0 ϕ dx + f j D j ϕ dx. Then u X is a (weak solution to problem (1 if ωa(x, u, u ϕ dx, B(u, ϕ = B 1 (u, ϕ + B 2 (u, ϕ = T (ϕ, for all ϕ X. Step 1. For j = 1,..., n we define the operator F j : X L p (, ω by (F j u(x = A j (x, u(x, u(x. We now show that operator F j is bounded and continuous.

7 38 Albo Carlos Cavalheiro (i Using (H4 we obtain F j u p L p (,ω = F j u(x p ω dx = A j (x, u, u p ω dx p (K 1 + h 1 u p/p + h 2 u p/p ω dx [ ] C p (K p 1 + hp 1 u p + h p 2 u p ω dx [ = C p K p 1 ω dx + h p 1 u p ω dx + h p 2 u p ω dx where the constant C p depends only on p. We have, by Theorem 3, h p 1 u p ω dx h 1 p L ( C p h 1 p L ( u p ω dx u p ω dx C p h 1 p L ( u p X, ], (3 and h p 2 u p ω dx h 2 p L ( u p ω dx h 2 p L ( u p X. Therefore, in (3 we obtain F j u L p (,ω C p ( K L p (,ω + (Cp/p h 1 L ( + h 2 L ( u p/p X, and hence the boundedness. (ii Let u m u in X as m. We need to show that F j u m F j u in L p (, ω. We will apply the Lebesgue Dominated Convergence Theorem. If u m u in X, then u m u in L p (, ω and u m u in L p (, ω. Using Theorem 2, there exist a subsequence {u mk } and functions Φ 1 and Φ 2 in L p (, ω such that u mk (x u(x, µ 1 - a.e. in, u mk (x Φ 1 (x, µ 1 - a.e. in, u mk (x u(x, µ 1 - a.e. in, u mk (x Φ 2 (x, µ 1 - a.e. in.

8 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 39 where µ 1 (E = ω(x dx. Hence, using (H4, we obtain E F j u mk F j u p = F L p (,ω j u mk (x F j u(x p ω dx = A j (x, u mk, u mk A j (x, u, u p ω dx C p ( A j (x, u mk, u mk p + A j (x, u, u p ω dx C p [ + 2C p (K 1 + h 1 u mk p/p + h 2 u mk p/p p (K 1 + h 1 u p/p + h 2 u p/p p 2C p [ ( K 1 + h 1 Φ p/p 1 + h 2 Φ p/p 2 K p 1 ω dx + p h p 1 Φp 1 ω dx + [ 2C p K 1 p + h L p (,ω 1 p L ( ] + h 2 p L ( Φ p 2 ω dx ] ω dx ω dx Φ p 1 ω dx 2C p [ K 1 p L p (,ω + h 1 p L ( Φ 1 p L p (,ω + h 2 p L ( Φ 2 p L p (,ω ]. ω dx h p 2 Φp 2 ω dx ] By condition (H1, we have F j u m (x = A j (x, u m (x, u m (x A j (x, u(x, u(x = F j u(x, as m +. Therefore, by the Dominated Convergence Theorem, we obtain F j u mk F j u L p (,ω 0, that is, F ju mk F j u in L p (, ω. By the Convergence principle in Banach spaces (see Proposition in [15] we have F j u m F j u in L p (, ω. (4 Step 2. We define the operator G: X L q (, v by (Gu(x = u(x q 2 u(x. We also have that the operator G is continuous and bounded. In fact,

9 40 Albo Carlos Cavalheiro (i We have Gu q L q (,v = = = u q 2 u q v dx u (q 2q u q v dx u q v dx u q X. Hence, Gu L q (,v u q/q X. (ii If u m u in X then u m u in L q (, v. By Theorem 2, there exist a subsequence {u mk } and a function Φ 3 L q (, v such that u mk (x u(x, µ 2 - a.e. in u mk (x Φ 3 (x, µ 2 - a.e. in. where µ 2 (E = v(x dx. Hence, using Lemma 1(a, we obtain, if q 2 E Gu mk Gu q = Gu L q (,v mk Gu q v dx = u m k q 2 u mk u q 2 q u v dx α q q [α q u mk u ( u mk + u (q 2 ] q α q q 2 (q 2q ( u mk u q (2Φ 3 (q 2q v dx ( Φ (q 2qq /(q q 3 v dx q /q u mk u q v dx (q q /q α q q 2 (q 2q u mk u q X Φ q q L q (,v, since (q 2qq /(q q = q if q 2. If q = 2, we have Gu mk Gu 2 L 2 (,v = u mk u 2 v dx u mk u 2 X. v dx Therefore (for 1 < q <, by the Dominated Convergence Theorem, we obtain Gu mk Gu L q (,v 0, that is, Gu mk Gu in L q (, v. By the Convergence principle in Banach spaces (see Proposition in [15], we have Gu m Gu in L q (, v. (5

10 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 41 Step 3. We have, by Theorem 3, T (ϕ = f 0 ϕ dx + f 0 n ω ϕ ω dx + f j D j ϕ dx f j ω D jϕ ω dx n f 0 /ω L p (,ω ϕ L p (,ω + f j /ω L p (,ω D jϕ Lp (,ω (C n f 0 /ω L p (,ω + f j /ω L p (,ω ϕ X. Moreover, using (H4 and the Hölder inequality, we also have and B(u, ϕ B 1 (u, ϕ + B 2 (u, ϕ A j (x, u, u D j ϕ ω dx + In (6 we have A(x, u, u ϕ ω dx u q 2 u ϕ v dx. (6 (K 1 + h 1 u p/p + h 2 u p/p ϕ ω dx K 1 L p (,ω ϕ L p (,ω + h 1 L ( u p/p L p (,ω ϕ L p (,ω + h 2 L ( u p/p L p (,ω ϕ L p (,ω ( K 1 L p (,ω + (Cp/p h 1 L ( + h 2 L ( u p/p X ϕ X, u q 2 u ϕ v dx = u q 1 ϕ v dx ( 1/q ( 1/q u q v dx ϕ q v dx u q/q X ϕ X. Hence, in (6 we obtain, for all u, ϕ X B(u, ϕ [ K 1 L p (,ω + Cp/p h 1 L ( u p/p X ] ϕ X. + h 2 L (,ω u p/p X + u q/q X Since B(u, is linear, for each u X, there exists a linear and continuous operator A: X X such that Au, ϕ = B(u, ϕ, for all u, ϕ X (where f, x

11 42 Albo Carlos Cavalheiro denotes the value of the linear functional f at the point x and Au K 1 L p (,ω + Cp/p + h 2 L (,ω u p/p X h 1 L ( u p/p X + u q/q X. Consequently, problem (1 is equivalent to the operator equation Au = T, u X. Step 4. Using condition (H2 and Lemma 1(b, we have Au 1 Au 2, u 1 u 2 = B(u 1, u 1 u 2 B(u 2, u 1 u 2 = ω A(x, u 1, u 1. (u 1 u 2 dx + u 1 q 2 u 1 (u 1 u 2 v dx ωa(x, u 2, u 2 (u 1 u 2 dx u 2 q 2 u 2 (u 1 u 2 v dx ( = ω A(x, u 1, u 1 A(x, u 2, u 2 (u 1 u 2 dx + ( u 1 q 2 u 1 u 2 q 2 u 2 (u 1 u 2 v dx θ 1 ω (u 1 u 2 p dx + β q ( u1 + u 2 q 2 u1 u 2 2 v dx θ 1 ω (u 1 u 2 p ( dx + β q u1 u 2 q 2 u1 u 2 2 v dx = θ 1 ω (u 1 u 2 p dx + β q u 1 u 2 q v dx 0. Therefore, the operator A is monotone. Moreover, using (H3, we obtain Au, u = B(u, u = B 1 (u, u + B 2 (u, u = ω A(x, u, u. u dx + u q 2 u u v dx λ 1 u p ω dx + u q v dx Hence, since 1 < p, q <, we have = λ 1 u p L p (,ω + u q L q (,v. Au, u u X +, as u X +, t p + s q (using lim = that is, A is coercive. t+s t + s Step 5. We need to show that the operator A is continuous.

12 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 43 Let u m u in X as m. We have, B 1 (u m, ϕ B 1 (u, ϕ A j (x, u m, u m A j (x, u, u D j ϕ ω dx = F j u m F j u D j ϕ ω dx F j u m F j u L p (,ω D jϕ Lp (,ω F j u m F j u L p (,ω ϕ X, and B 2 (u m, ϕ B 2 (u, ϕ = u m q 2 u m ϕv dx u q 2 u ϕv dx u m q 2 u m u q 2 u ϕ v dx = Gu m Gu ϕ v dx for all ϕ X. Hence, Gu m Gu L q (,v ϕ L q (,v Gu m Gu L q (,v ϕ X, B(u m, ϕ B(u, ϕ B 1 (u m, ϕ B 1 (u, ϕ + B 2 (u m, ϕ B 2 (u, ϕ [ n ] F j u m F j u L p (,ω + Gu m Gu L q (,v ϕ X. Then we obtain Au m Au F j u m F j u L p (,ω + Gu m Gu L q (,v. Therefore, using (4 and (5 we have Au m Au 0 as m +, that is, A is continuous (and this implies that A is hemicontinuous. Therefore, by Theorem 4, the operator equation Au = T has a solution u X and it is a solution for problem (1. Step 6. Let us now prove the uniqueness of the solution. Suppose that u 1, u 2 X are two solutions of problem (1. Then, u i q 2 u i ϕ v dx + ω A j (x, u i (x, u i (xd j ϕ dx = f 0 ϕ dx + f j D j ϕ dx,

13 44 Albo Carlos Cavalheiro for all ϕ X, and i = 1, 2. Hence, we obtain ( u 1 q 2 u 1 u 2 q 2 u 2 ϕv dx ( + ω A(x, u 1 (x, u 1 (x A(x, u 2 (x, u 2 (x ϕ dx = 0. In particular, for ϕ = u 1 u 2 X we have, by (H2 and Lemma 1(b (analogous to Step 4, ( 0 = ω A(x, u 1, u 1 A(x, u 2, u 2 (u 1 u 2 dx + ( u 1 q 2 u 1 u 2 q 2 u 2 (u 1 u 2 v dx θ 1 (u 1 u 2 p ω dx + β q (u 1 u 2 q v dx. Hence, (u 1 u 2 Lp (,ω = 0 and (u 1 u 2 Lq (,v = 0. Since u 1, u 2 X, then u 1 = u 2 µ 1 -a.e. Therefore, since ω A p, we obtain that u 1 = u 2 a.e. Example 1. Consider = {(x, y R 2 : x 2 + y 2 < 1}, the weight functions ω(x, y = (x 2 + y 2 1/2 and v(x, y = (x 2 + y 2 2/3 (ω A 3 and v A 2, p = 3, q = 2, and the function A: R 2 R 2 A((x, y, η, ξ = h 2 (x, y ξ ξ, where h(x, y = 2 e (x2 +y 2. Let us consider the partial differential operator Lu(x, y = ( (x 2 + y 2 2/3 u u div ( (x 2 + y 2 1/2 A((x, y, u, u. Therefore, by Theorem 1, the problem (1 Lu(x = cos(xy (x 2 + y 2 ( sin(xy x (x 2 + y 2 ( sin(xy y (x 2 + y 2, in u(x = 0, u(x = 0, has a unique solution u X = W 2,2 (, v W 1,3 0 (, ω. References on on [1] A.C. Cavalheiro: Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems. J. Appl. Anal. 19 ( [2] A.C. Cavalheiro: Existence results for Dirichlet problems with degenerate p-laplacian. Opuscula Math. 33 (3 (

14 Existence of solutions for Navier problems with degenerate nonlinear elliptic equations 45 [3] M. Chipot: Elliptic Equations: An Introductory Course. Birkhäuser, Berlin (2009. [4] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities. Walter de Gruyter, Berlin (1997. [5] E. Fabes, C. Kenig, R. Serapioni: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7( [6] S. Fučík, O. John, A. Kufner: Function Spaces. Noordhoff International Publ., Leiden (1977. [7] J. Garcia-Cuerva, J.L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies (1985. [8] D. Gilbarg, N.S. Trudinger: Elliptic Partial Equations of Second Order. Springer, New York (1983. [9] J. Heinonen, T. Kilpeläinen, O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math., Clarendon Press (1993. [10] A. Kufner: Weighted Sobolev Spaces. John Wiley & Sons (1985. [11] B. Muckenhoupt: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 ( [12] M. Talbi, N. Tsouli: On the spectrum of the weighted p-biharmonic operator with weight. Mediterr. J. Math. 4 ( [13] A. Torchinsky: Real-Variable Methods in Harmonic Analysis. Academic Press, São Diego (1986. [14] B.O. Turesson: Nonlinear Potential Theory and Weighted Sobolev Spaces. Springer-Verlag (2000. [15] E. Zeidler: Nonlinear Functional Analysis and its Applications, Vol. I. Springer-Verlag (1990. [16] E. Zeidler: Nonlinear Functional Analysis and its Applications, Vol. II/B. Springer-Verlag (1990. Author s address: Department of Mathematics, State University of Londrina, Londrina, Brazil accava a gmail.com Received: 16th September, 2014 Accepted for publication: 8th February, 2015 Communicated by: Geoff Prince