Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

Abstract In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L1-right-hand sides in a bounded domain of ℝn(n ⩾ 2) . This class is described by the presence of a set of exponents q1,…, qn and a set of weighted functions ν1,…, νn in growth and coercitivity conditions on coefficients of the equations. The exponents qi characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions νi characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.


Introduction
During the last twenty years the research on the existence and properties of solutions for nonlinear equations and variational inequalities with L 1 -data or measures as data were intensively developed. As is generally known, an effective approach to the solvability of second-order equations in divergence form with L 1 -right-hand sides has been proposed in [1]. In this connection we also mention a series of other close investigations for nondegenerate isotropic nonlinear second-order equations with L 1 -data and measures, entropy and renormalized solutions [2][3][4][5][6][7][8][9][10].
As for the solvability of nonlinear elliptic second-order equations with anisotropy and degeneracy (with respect to the independent variables), we note the following works. The existence of a weak (distributional) solution to the Dirichlet problem for a model nondegenerate anisotropic equation with right-hand side measure was established in [11]. The existence of weak solutions for a class of nondegenerate anisotropic equations with locally integrable data in R n .n > 2/ was proved in [12], and an analogous existence result concerning the Dirichlet problem for a system of nondegenerate anisotropic equations with measure data was obtained in [13]. Moreover, in [14], the existence of weak solutions to the Dirichlet problem for nondegenerate anisotropic equations with right-hand sides from Lebesgues spaces close to L 1 was established. Solvability of the Dirichlet problem for degenerate isotropic equations with L 1 -data and measures as data was studied in [15][16][17][18][19]. Remark that in [15,17], the existence of entropy solutions to the given problem was proved in the case of L 1 -data, and in [16], the existence of a renormalized solution of the problem for the same case was established. In [16,18,19], the existence of distributional solutions of the problem was obtained in the case of right-hand side measures.
Solvability of the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L 1 -right-hand sides was studied in [20]. This class is described by the presence of a set of exponents q 1 ; : : : ; q n and of a set of weighted functions 1 ; : : : ; n in growth and coercitivity conditions on coefficients of the equations under *Corresponding Author: Yuliya Gorban: Donetsk National University, Vinnytsya, Ukraine, E-mail: yuliya_gorban@mail.ru consideration. The exponents q i characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions i characterize degeneration or singularity of the coefficients with respect to the independent variables. This is the most general situation in comparison with the above-mentioned works: the nondegenerate isotropic case means that i Á 1 and q i D q 1 , i D 1; : : : ; n; the nondegenerate anisotropic case means that i Á 1, i D 1; : : : ; n, and q i , i D 1; : : : ; n, are generally different, and the degenerate isotropic case means that i D 1 , i D 1; : : : ; n, as in [16][17][18][19] or i , i D 1; : : : ; n, are generally different as in [15] but q i D q 1 , i D 1; : : : ; n.
In [20], the theorem on the existence and uniqueness of entropy solution to the Dirichlet problem for this class of the equations was proved. Moreover, the existence results of some other types of solutions to the given problem were also obtained. Observe that the proofs of these theorems are based on use of some results of [21][22][23] on the existence and properties of solutions of second-order variational inequalities with L 1 -right-hand sides and sufficiently general constraints. Note that in [20][21][22][23] right-hand sides to the investigated variational inequalities and equations depend on independent variables only, and belong to the class L 1 .
The present article is devoted to the Dirichlet problem for a same class of the nonlinear elliptic second-order equations in divergence form with degenerate anisotropic coefficients as in [20]. Here right-hand sides to the given equations depend on independent variables and unknown function. A model example of this class is an equation where is a bounded domain in R n .n > 2/, 1 < q i < n, i > 0 a.e. in , i 2 L 1 loc . /, .1= i / 1=.q i 1/ 2 L 1 . /, i D 1; : : : ; n, F W R ! R is a Carathéodory function. The main result of this paper is a theorem on the existence of entropy solutions to the Dirichlet problem for the equations under consideration. We require an additional conditions to the function F in this theorem. Namely F .x; u/ has an arbitrary growth with respect to the second variable, and F .x; u/ belongs to L 1 . / under the fixed value of the second variable. In our case we have no opportunity to use the results [21][22][23] directly. We follow a general approach for proving the above-mentioned theorem. This approach has been proposed in [1] to the investigation on the existence and properties of solutions for nonlinear elliptic second-order equations with isotropic nondegenerate (with respect to the independent variables) coefficients and L 1 -right-hand sides. In [21,23] this approach has been taken to the anisotropic degenerate case. Also we use some ideas of [24].
q D min fq i W i D 1; : : : ; ng; q D 1 n Let for every i 2 f1; : : : ; ng i be a nonnegative function on such that i > 0 a.e. in , We set D f i W i D 1; : : : ; ng. We denote by W 1;q . ; / the set of all functions u 2 W 1;1 . / such that for every i 2 f1; : : : ; ng we have i j D i u j q i 2 L 1 . /. Let k k 1;q; be the mapping from W 1;q . ; / into R such that for every function u 2 W 1;q . ; / The mapping k k 1;q; is a norm in W 1;q . ; /, and, in view of the second inclusion of (1), the set W 1;q . ; / is a Banach space with respect to the norm k k 1;q; . Moreover, by virtue of the first inclusion of (1), we have C 1 0 . / W 1;q . ; /. We denote by ı W 1;q . ; / the closure of the set C 1 0 . / in space W 1;q . ; /. Evidently, the set ı W 1;q . ; / is a Banach space with respect to the norm induced by the norm k k 1;q; . It is obvious that ı W 1;q . ; / ı W 1;1 . /.
Finally, we observe that ı W 1;q . ; / is a reflexive space. The proof of the latter statement can be found in [21]. Note that the following assertion hold. Further, let for every k > 0 T k W R ! R be the function such that By analogy with known results for nonweighted Sobolev spaces (see for instance [25]) we have: if u 2 ı W 1;q . ; / and k > 0, then T k .u/ 2 ı W 1;q . ; / and for every i 2 f1; : : : ; ng We denote by ı T 1;q . ; / the set of all functions u W ! R such that for every k > 0, T k .u/ 2 ı W 1;q . ; /. Clearly, For every u W ! R and for every x 2 we set k.u; x/ D minfl 2 N W ju.x/j 6 lg: Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

3 Statement of the Dirichlet problem. The concept of its entropy solution
Let c 1 , c 2 > 0, g 1 , g 2 2 L 1 . /, g 1 , g 2 > 0 in , and let for every i 2 f1; : : : ; ng a i W R n ! R be a Carathéodory function. We suppose that for almost every x 2 and for every 2 R n , Moreover, we assume that for almost every x 2 and for every ; Now we give one result of [20] which will be used in the sequel.
We consider the following Dirichlet problem: u D 0 on @ : Definition 3.2. An entropy solution of problem (7), (8) is a function u 2 ı T 1;q . ; / such that: for every function w 2 ı W 1;q . ; / \ L 1 . / and for every k > 1 Note that the left-hand integral in (10) is finite. It follows from assertion b) of Proposition 3.1. The right-hand integral in (10) is also finite. It follows from the boundedness of the function T k and inclusion (9).

Main result
Next theorem is the main result of this paper. 2) for any s 2 R the function F . ; s/ belongs to L 1 . /.
Then there exists an entropy solution of the Dirichlet problem (7), (8).
Proof. According to the approach from [1], we will consider a sequence of the approximating problems for the equations with smooth right-hand sides. Then we will obtain special estimates of the solutions of these problems. Finally, we will pass to the limit. The proof is in 9 steps.
Step 1. We set f D F . ; 0/. Let for every l 2 N, F l W R ! R be the function such that By virtue of condition 1), we have: if l 2 N; then for a.e. x 2 the function F l .x; / is nondecreasing on R: Further, in view of condition 2), we have f 2 L 1 . /. Hence there exists ff l g C 1 0 . / such that: Using the inequalities (4)-(6), property (11), and well-known results on the solvability of the equations with monotone operators (see for instance [26]), we obtain: if l 2 N, then there exists the function u l 2 ı W 1;q . ; / such that for every function w 2 It means that the function u l 2 ı W 1;q . ; / is a generalized solution of the Dirichlet problem: We denote by c i , i D 3; 4; : : : , the positive constants depending only on n, q, c 1 , c 2 , kg 1 k L 1 . / , kg 2 k L 1 . / , kf k L 1 . / , kF . ; 1/k L 1 . / , kF . ; 1/k L 1 . / , k1= i k L 1=.q i 1/ . / , i D 1; : : : ; n, and meas . Let us show that for every k > 1 and l 2 N the following inequalities hold: In fact, let k > 1 and l 2 N. As u l 2 ı W 1;q . ; /, we have T k .u l / 2 ı W 1;q . ; /. In view of (14) and (13) Using (2) and (5) in the left-hand side of this inequality, we get Assertion (11) and properties of the function T k imply that The estimate (15) follows from (18) and (17). Finally, the inequality (16) follows from (19) and (17).
Step 2. Now we show that for every k > 1 and l 2 N In fact, let k > 1 and l 2 N. We have jT k .u l /j D k on fju l j > kg; then Using Proposition 2.6, (2) and (15), we obtain The inequality (20) follows from the latter estimate and (22). Next, we fix i 2 f1; : : : ; ng, and set We have meas f 1=q i i jD i u l j > kg 6 meas fju l j > k g C meas G: From (20) it follows that meas fju l j > k g 6 c 5 k O q : Moreover, in view of the set's G definition and (15) we get k q i meas G 6 Z fju l j<k g i jD i u l j q i dx 6 c 3 k : The inequality (21) follows from the latter estimate and (23), (24).
Step 3. Assertions (2) and (15) imply that for every k > 1 the sequence fT k .u l /g is bounded in ı W 1;q . ; /. As the space ı W 1;q . ; / is reflexive, then there exist an increasing sequence fl h g N, and sequence fz k g ı W 1;q . ; / such that for every k 2 N we have a weak convergence T k .u l h / ! z k in ı W 1;q . ; /. Without loss of generality it can be assumed that Step 4. Let us show that the sequence fu l g is fundamental on measure. Indeed, let k > 1, l; j 2 N. We fix t > 0, and set G 0 D fju l j < k; ju j j < k; ju l u j j > t g. It is clear that meas fju l u j j > t g 6 meas fju l j > kg C meas fju j j > kg C meas G 0 : As t 6 jT k .u l / T k .u j /j on G 0 , we obtain t meas G 0 6 Z jT k .u l / T k .u j /j dx: This inequality, (20) and (26) imply that for every k > 1, and l; j 2 N meas fju l u j j > tg 6 2c Let " > 0. We fix k 2 N such that 2c 5 k O q 6 "=2: Taking into account (25) and Proposition 2.1, we infer a strong convergence T k .u l / ! z k in L 1 . /. Then there exists N 2 N such that for every l; j 2 N, l; j > N Z jT k .u l / T k .u j /j dx 6 "t =2: From this inequality, (27), and (28) we deduce that for every l; j 2 N, l; j > N meas fju l u j j > tg 6 ": This means that the sequence fu l g is fundamental on measure.
Step 5. Now we show that for every i 2 f1; : : : ; ng the sequence f 1=q i i D i u l g is fundamental on measure. For every t > 0 and l; j 2 N we put Besides, for every t > 0, h; k > 1, and l; j 2 N we set : Using (21), we establish that for every t > 0, h > n, k > 1, and l; j 2 N N t .l; j / 6 2c 6 n nC1 h q O q=.1C O q/ C meas fju l u j j > 1=kg C meas E t;h;k .l; j /: Further, we get one estimate for some integrals over E t;h;k .l; j /. So we introduce now auxiliary functions and sets. Let for every x 2 ˆx W R n R n ! R be a function such that for every pair . ; 0 / 2 R n R n Recall that a i ; i D 1; : : : ; n, are Carathéodory functions, and inequality (6) holds for almost every x 2 and every ; 0 2 R n , 6 D 0 . Then there exists a set E , meas E D 0, such that: (i) for every x 2 nE the functionˆx is continuous on R n R n ; (ii) for every x 2 nE and ; 0 2 R n ; 6 D 0 , we haveˆx. ; 0 / > 0.
Put for every t > 0, h > t, and x 2 As i > 0 a.e. in ; i D 1; : : : ; n, then there exists a set Q E , meas Q E D 0, such that the set G t;h .x/ is nonempty for every t > 0; h > t , and x 2 n Q E.
Let for every t > 0 and h > t t;h W ! R be a function such that For every t > 0 and h > t we have t;h > 0 a.e. in , and t;h 2 L 1 . /. Let t > 0, h > t C1, k > 1, and l; j 2 N. We fix x 2 E t;h;k .l; j /n.E [ Q E/, and set D ru l .x/, 0 D ru j .x/. As . ; 0 / 2 G t;h .x/, then t;h .x/ 6ˆx. ; 0 /. This inequality and function'sˆx definition imply that Then, taking into account (6) and (2), we obtain In view of (14) we have

From these equalities and (30) it follows that
Using (16) and conditions 1), 2), we find that for every l; j 2 N Z jF l .x; u l / F j .x; u j /j dx 6 c 7 : From the latter estimate and (31) we deduce that for every t > 0; h > t C 1; k > 1, and l; j 2 N the following inequality holds: The sequence fu l g is fundamental on measure. Then there exists an increasing sequence fn k g N such that for every k 2 N and l; j 2 N, l; j > n k , we have meas fju l u j j > 1=kg 6 1=k: Let t > 0 and " > 0. We fix h > t C n such that Put for every k 2 N˛k D sup l;j >n k meas E t;h;k .l; j /: Let us show that˛k ! 0. Assume the converse. Then there exist > 0, an increasing sequence fk s g N, and sequences fl s g; fj s g N such that for every s 2 N we have l s ; j s > n k s and meas E t;h;k s .l s ; j s / > : Let G s D E t;h;k s .l s ; j s /, s 2 N.
In view of (32) and (13)  From this assertion, taking into account t;h 2 L 1 . / and t;h > 0 a.e. in , we infer that meas G s ! 0. This fact is in contradiction to (35). Hence, we conclude that˛k ! 0. Finally, we fix k 2 N such that the inequalities hold: Let l; j 2 N; l; j > n k . From (29), (33), (34) and (36) it follows that N t .l; j / 6 ". This means that for every i 2 f1; : : : ; ng the sequence f 1=q i i D i u l g is fundamental on measure.
Step 6. From results of the Steps 4 and 5, and F. Riesz's theorem we get the following facts: there exist measurable functions u W ! R and v .i / W ! R; i D 1; : : : ; n; such that the sequence fu l g converges to u on measure, and for every i 2 f1; : : : ; ng the sequence f 1=q i i D i u l g converges to v .i/ on measure. As is generally known, we can extract subsequences converging almost everywhere in to the corresponding functions. We may assume without loss of generality that u l ! u a.e. in ; (37) 8 i 2 f1; : : : ; ng Step 7. Now we show that 8 i 2 f1; : : : ; ng D i u l ! ı i u a.e. in : In fact, let i 2 f1; : : : ; ng. In view of (37) there exists a set E 0 , meas E 0 D 0, such that and in view of (38) there exists a set E 00 , meas E 00 D 0, such that 8 x 2 nE 00 Fix k 2 N. By (2) we have: if l 2 N, then there exists a set E .l/ , meas E .l/ D 0, such that We denote by O E a union of sets E 0 ; E 00 and E .l/ , l 2 N. Clearly, meas O E D 0: Let x 2 fjuj < kgn O E. In view of (42) there exists l 0 2 N such that for every l 2 N; l > l 0 , we have ju l .x/j < k. Let l 2 N; l > l 0 . Then a.e. in fjuj < kg: Besides, in view of (2) and (15) for every l 2 N Using Fatou's lemma, from (45) and (46) we infer that the function jv .i/ j q i is summable in fjuj < kg. Further, let ' W ! R be a measurable function such that j'j 6 1 in , and let " > 0. As the function jv .i/ j is summable on fjuj < kg, then there exists " 1 2 .0; "/ such that for every measurable set G fjuj < kg; meas G 6 Moreover, in view of (45) and Egorov's theorem, there exists a measurable set 0 fjuj < kg such that meas .fjuj < kgn 0 / 6 " 1 ; From (47) and (48) we infer that Z and from (49) we deduce that there exists l 1 2 N such that for every l 2 N; l > l 1 , Let l 2 N; l > l 1 . Using (50), (51), Hölder's inequality, (48), and (46), we geť Since " is an arbitrary constant, from the latter estimate it follows that On the other hand, let F W ı W 1;q . ; / ! R be a functional such that for every function v 2 It is easy to see that F 2 .
From (52) and (53) we deduce that In turn, from this equality and Proposition 2.1 we infer that v .i/ D 1=q i i ı i u a.e. in fjuj < kg: Since k 2 N is an arbitrary number, from the latter assertion it follows that Taking into account that i > 0 a.e. in , from (38) and (54) Step 8. Let us show that the following assertions are fulfilled: Indeed, in view of (37) we have F l .x; u l / ! f F .x; u/ a.e. in : Moreover, using (16) and conditions 1) and 2), we get for every l 2 N Z jF l .x; u l /j dx 6 c 8 : From this fact, (58), and Fatou's lemma we obtain inclusion (56). Now let us prove (57). Firstly, we establish that for every k, l 2 N the following estimate holds Let z 2 C 1 .R/ be a function such that 0 6 z 6 1 on R, z D 0 on OE 1I 1, z D 1 on . 1I 2 [ OE2I C1/, and for every s 2 R z 0 .s/sign s > 0, jz 0 .s/j 6 2.
We fix arbitrary k, l 2 N. We denote by z k W R ! R a function such that for every s 2 R From the properties of the functions T 1 and z it follows that for every s 2 R jz k .s/j 6 1: Besides, 8s 2 R; jsj 6 k; z k .s/ D 0I (62) 8s 2 R; jsj > 2k; jz k .s/j D 1: Definition (60) implies that z k .u l / 2 ı W 1;q . ; / and e. in ; i D 1; : : : ; n: Substituting w D z k .u l / into (14), and using (61), (62), we get We denote by I 0 k;l the first integral in the left-hand side of (65). In view of the function's z definition 8s 2 R; jsj 6 k; or jsj > 2k; jz 0 .s/j D 0: Using (64), (66), and (5), we establish that From the truncated function's property and our condition z 0 .s/sign s > 0, 8s 2 R, it follows that almost everywhere in fk 6 ju l j 6 2kg z 0 u l k Taking into account this fact and our condition jz 0 .s/j 6 2, 8s 2 R, we deduce from (67) This and (65) imply Note that in view of (11) and the function's z k definition we have and in view of (63) we get F l .x; u l /z k .u l / D jF l .x; u l /j a.e. in fju l j > 2kg: jF l .x; u l /j dx: Finally, assertion (59) is derived from the latter inequality and (68). Next, we fix an arbitrary " > 0. It is clear that there exists " 1 > 0 such that for every measurable set G , meas G 6 " 1 , Z G .jf j C jF .x; u/j/ dx 6 ": We fix k 2 N such that the following inequalities hold: By condition 2), we infer that the functions F . ; 2k/ and F . ; 2k/ belong to L 1 . /. Hence, there exists " 2 > 0 such that for every measurable set G , meas G 6 " 2 , Z G .jF . ; 2k/j C jF . ; 2k/j/ dx 6 ": In view of (58) there exists a measurable set 1 such that meas . n 1 / 6 min." 1 ; " 2 /; and F l .x; u l / ! f F .x; u/ uniformly in 1 . Then there exists L 1 2 N such that for every l 2 N, l > L 1 , Besides, in view of (12) there exists L 2 2 N such that for every l 2 N, l > L 2 , Note that by condition 1), we have jF l .x; u l /j 6 2.jF .x; 2k/j C jF .x; 2k/j/ a.e. in fju l j < 2kg: In view of (71) we get Z n 1 .jF .x; 2k/j C jF .x; 2k/j/ dx 6 ": Assertion (94) means that we can find l 0 2 N such that for every l 2 N; l > l 0 ; and x 2 1 ju l .x/ u.x/j 6 ": Moreover, in view of (95) there exists l 1 2 N such that for every l 2 N, l > l 1 , we get Taking into account Hölder inequality, (4), an inclusion H l n O E fju l j < k 1 g, (15), and (93), we established that for every i 2 f1; : : : ; ng Z H l n 1 ja i .x; ru l /j jD i wj dx 6 .c 1 c 3 k 1 C 1 C kg 1 k L 1 . / /": On the other hand, for almost every x 2 H 0 the inequality ju.x/j < k 1 C 1 holds. So, T k .u w/ D T k 1 C1 .u/ w a.e. in H 0 . Therefore, D i T k .u w/ D D i T k 1 C1 .u/ D i w a.e. in H 0 : Then, taking into account (100), we get D i T k 1 C1 .u/ D D i w a.e. in H 0 . This and Proposition 2.4 imply that ı i u D D i w a.e. in H 0 . From this result and (41) we infer that for every i 2 f1; : : : ; ng D i u l ! D i w a.e. in H 0 . Hence, Using (5), (88)-(92), (99), Fatou's lemma, and (106), we established that the function n P iD1 a i .x; ıu/ı i u C g 2 ! is summable in and Z n X iD1 a i .x; ıu/ı i u C g 2 dx 6 Z F .x; u/ T k .u w/ dx C Z H .' 1 C g 2 / dx: From the latter inequality and Propositions 2.4 and 2.5 we obtain (80).