ISRN.MATHEMATICAL.ANALYSIS ISRN Mathematical Analysis 2090-4665 Hindawi Publishing Corporation 875145 10.1155/2014/875145 875145 Research Article A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space Kar Rasmita Colombini F. Marusic-Paloka E. Mascia C. TIFR Center for Applicable Mathematics Karnataka Bangalore 560065 India 2014 2532014 2014 09 12 2013 12 02 2014 26 3 2014 2014 Copyright © 2014 Rasmita Kar. 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.

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem L u - μ u g 1 + h u , u g 2 = f in Ω , u = 0 on Ω , in a suitable weighted Sobolev space, where Ω n is a bounded domain and h is a continuous bounded nonlinearity.

1. Introduction

Let Ω n be a bounded domain with boundary Ω . Let L be an operator in divergence form: (1) L u ( x ) = - i , j = 1 n D j ( a i j ( x ) D i u ( x ) ) with    D j = x j , with coefficients a i j / ω L ( Ω ) which are symmetric and satisfy the degenerate ellipticity condition: (2) λ | ξ | 2 ω ( x ) i , j = 1 n a i j ( x ) ξ i ξ j Λ | ξ | 2 ω ( x ) , a . e .    x Ω , for all ξ n , and ω is an A 2 -weight ( λ > 0 , Λ > 0 ) . Let μ and f / ω L 2 ( Ω , ω ) and let h be a real valued continuous function defined on × n . In this paper, we study the existence of weak solution of the BVP: (3) L u - μ u g 1 + h ( u , u ) g 2 = f in    Ω , u = 0 on    Ω , under suitable hypotheses on the functions g 1 , g 2 , and h . The present work is inspired by a semilinear problem in bounded domain given in the book by Zeidler . In general, the Sobolev spaces W k , p ( Ω ) without weights occur as spaces of solutions for elliptic and parabolic PDEs. For degenerate problems with various types of singularities in the coefficients it is natural to look for solutions in weighted Sobolev spaces; for example, see . Section 2 deals with preliminaries and some basic results. Section 3 contains the main result and is about the existence of a weak solution to (3) in a suitable weighted Sobolev space.

2. Preliminaries

We need the following preliminaries for the ensuing study. Let Ω n be a bounded domain (open connected set). Let ω : n + be a locally integrable function with 0 < ω < a.e. 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 constant c = c p , ω such that (4) ( 1 | B | B ω ( x ) d x ) ( 1 | B | B ω 1 / ( 1 - p ) ( x ) d x ) p - 1 c , for all balls B in n , where | · | denotes the n -dimensional Lebesgue measure in n . We assume that ω A p , 1 < p < . We will denote by L p ( Ω , ω ) ( 1 p < ) the usual Banach space of measurable real valued functions, f , defined in Ω for which (5) f p , Ω = ( Ω | f ( x ) | p ω ( x ) d x ) 1 / p < . For p 1 and a positive integer k , the weighted Sobolev space W k , p ( Ω , ω ) is defined by (6) W k , p ( Ω , ω ) : = { u L p ( Ω , ω ) : D α u L p ( Ω , ω ) , 1 | α | k } with the associated norm (7) u k , p , Ω = u p , Ω + 1 | α | k D α u p , Ω . In order to avoid too many suffices, at each step, a generic constant is denoted by c or C Ω . We need the following result.

Proposition 1 (the weighted Sobolev inequality).

Let Ω n be a bounded domain and let ω A p ( 1 < p < ) . Then, there exist positive constants C Ω and δ such that, for all u C 0 ( Ω ) and all k satisfying 1 k n / ( n - 1 ) + δ , (8) u k p , Ω C Ω u p , Ω .

A proof of the above statement can be found in [5, Theorem 1.3].

For p = 2 and k = 1 in the above inequality, we have (9) u 2 , Ω C Ω u 0,1 , 2 , where (10) u 0,1 , 2 = u 2 , Ω : = ( Ω | u ( x ) | 2 ω d x ) 1 / 2 . Further, we use function space W 0 1,2 ( Ω , ω ) which is defined as the closure of C 0 ( Ω ) with respect to the norm u 0,1 , 2 (correctness of definition of this norm follows from inequality (9)). We also note that W 1,2 ( Ω , ω ) and W 0 1,2 ( Ω , ω ) are Hilbert spaces.

For more details on A p -weight and weighted Sobolev spaces, we refer to [5, 7, 911].

Proposition 2.

Let ω A p , 1 < p < , and let Ω be a bounded open set in n . If u n u in L p ( Ω , ω ) , then there exist a subsequence { u n k } and a function v L p ( Ω , ω ) such that

u n k u ( x ) as n k , ω -a.e. on Ω ;

| u n k ( x ) | v ( x ) , ω -a.e. on Ω .

Proof.

The proof of this theorem follows in the lines of Theorem 2.8 . 1 in .

Let ( f x ) denote the value of linear functional f at x .

Definition 3.

Let C , D : X X * be the operators on the real separable reflexive Banach space X . Then,

C satisfies condition ( S ) if (11) u n u ,    lim n ( C u n - C u u n - u ) = 0 , hhhhhhihhhhhhhhhhh implies    u n u ,

C is demicontinuous if and only if u n u as n implies C u n C u as n ,

C + D is asymptotically linear if C is linear and (12) D u u 0 , as u .

In Section 3, we use the following result.

Proposition 4.

Let B , N : X X * be operators on the real separable reflexive Banach space X . Then,

the operator B : X X * is linear and continuous;

the operator N : X X * is demicontinuous and bounded;

B + N is asymptotically linear;

for each T X * and for each t [ 0,1 ] , the operator A t defined by A t ( u ) : = B u + t ( N u - T ) satisfies condition ( S ) in X .

If B u = 0 implies u = 0 , then, for each T X * , the equation B u + N u = T has a solution in X .

For a detailed proof of the above theorem, we refer to  or to [1, Theorem 29 . C ].

Definition 5.

One says that u W 0 1,2 ( Ω , ω ) is a weak solution of (3) if (13) Ω a i j D i u ( x ) D j ϕ ( x ) d x - Ω μ u ( x ) g 1 ( x ) ϕ ( x ) d x + Ω h ( u ( x ) , u ( x ) ) g 2 ( x ) ϕ ( x ) d x = Ω f ( x ) ϕ ( x ) d x . for every ϕ W 0 1,2 ( Ω , ω ) .

We need the following hypotheses for further study.

( H 1 ) Let ( η , ξ ) h ( η , ξ ) be continuous in × n , where h is a bounded function (i.e., for a constant A > 0 , let | h ( t ) | A , t ).

( H 2 ) Let ω A 2 . Assume g 1 / ω L ( Ω ) , g 2 / ω L 2 ( Ω , ω ) , and f / ω L 2 ( Ω , ω ) .

( H 3 ) Assume g 2 ( h ( η , ξ ) - h ( η , ξ ) ) ( η - η ) 0 , where η , η and ξ , ξ n .

We define the functionals B 1 , B 2 : W 0 1,2 ( Ω , ω ) × W 0 1,2 ( Ω , ω ) by (14) B 1 ( u , ϕ ) = Ω a i j D i u ( x ) D j ϕ ( x ) d x hhhhhhhh - Ω μ u ( x ) g 1 ( x ) ϕ ( x ) d x , B 2 ( u , ϕ ) = Ω h ( u ( x ) , u ( x ) ) g 2 ( x ) ϕ ( x ) d x . Also define T : W 0 1,2 ( Ω , ω ) by (15) T ( ϕ ) = Ω f ( x ) ϕ ( x ) d x . A function u W 0 1,2 ( Ω , ω ) is a weak solution of (3) if (16) B 1 ( u , ϕ ) + B 2 ( u , ϕ ) = T ( ϕ ) , ϕ W 0 1,2 ( Ω , ω ) . By noting | a i j ( x ) | c ω ( x ) and by Hölder’s inequality, we get (17) | B 1 ( u , ϕ ) | Ω | a i j ( x ) | | D i u ( x ) | | D j ϕ ( x ) | d x i i i i i i i i i i i i i i i i + | μ | Ω | u ( x ) | | ϕ ( x ) | | g 1 ( x ) | d x i i i i i i i i i i i i i c Ω i , j = 1 n | D i u ( x ) | | D j ϕ ( x ) | ω ( x ) d x i i i i i i i i i i i i i i i i + | μ | Ω | u ( x ) | | ϕ ( x ) | | g 1 ( x ) ω ( x ) | ω ( x ) d x i i i i i i i i i i i i c ( Ω i = 1 n | D i u ( x ) | 2 ω ( x ) d x ) 1 / 2 i i i i i i i i i i i i i i i i × ( Ω j = 1 n | D j ϕ ( x ) | 2 ω ( x ) d x ) 1 / 2 i i i i i i i i i i i i i i i i + | μ | g 1 ω ( Ω | u ( x ) | 2 ω ( x ) d x ) 1 / 2 i i i i i i i i i i i i i i i i × ( Ω | ϕ ( x ) | 2 ω ( x ) d x ) 1 / 2 i i i i i i i i i i ( c + C Ω | μ | g 1 ω ) u 0,1 , 2 ϕ 0,1 , 2 , where C Ω is a constant arising out of (9). Now, B 1 ( · , · ) is linear and bounded. Then, there exists an operator (18) B : W 0 1,2 ( Ω , ω ) [ W 0 1,2 ( Ω , ω ) ] * , defined by ( B u    ϕ ) = B 1 ( u , ϕ ) , for all u , ϕ W 0 1,2 ( Ω , ω ) . Also, by ( H 1 ) and ( H 2 ) , it follows from Hölder’s inequality that (19) | B 2 ( u , ϕ ) | = | Ω g 2 ( x ) h ( u ( x ) , u ( x ) ) ϕ ( x ) d x | A Ω | g 2 ( x ) ω ( x ) | | ϕ ( x ) | ω ( x ) d x A    g 2 ω 2 , Ω ϕ 2 , Ω A    g 2 ω 2 , Ω ϕ 0,1 , 2 , and hence by the weighted Sobolev inequality (9) (20) | B 2 ( u , ϕ ) | C Ω | u | 0,1 , 2 | ϕ | 0,1 , 2 , u , ϕ W 0 1,2 ( Ω , ω ) . Now B 2 ( u , · ) is linear and so there exists an operator N : W 0 1,2 ( Ω , ω ) [ W 0 1,2 ( Ω , ω ) ] * such that (21) ( N u ϕ ) = B 2 ( u , ϕ ) , u , ϕ W 0 1,2 ( Ω , ω ) . Further, we have (22) | T ( ϕ ) |    Ω | f ( x ) | | ϕ ( x ) | d x f ω 2 , Ω ϕ 2 , Ω C Ω f ω 2 , Ω | ϕ | 0,1 , 2 . Then, problem (3) is equivalent to solving the operator equation (23) B u + N u = T , u W 0 1,2 ( Ω , ω ) .

3. Main Results

The main result of this section is to establish the existence of a solution for the degenerate nonlinear elliptic BVP (3), when μ > 0 is not an eigenvalue of (24) L u - μ u ( x ) g 1 ( x ) = 0 in    Ω , u = 0 on Ω , with certain restrictions. Also, two results are established related to the cases when g 1 does not change sign.

Theorem 6.

Assume the hypotheses ( H 1 ) ( H 3 ) and the inequality (25) λ > μ C Ω g 1 ω , where C Ω is a constant arising out of (9). Let μ > 0 not be an eigenvalue of (24). Then, the BVP (3) has a solution u W 0 1,2 ( Ω , ω ) .

Proof.

Idea of proof is such. First we write a weak solution of the BVP (3) as solution of operator equation (26) u W 0 1,2 ( Ω , ω ) : B u + N u = T in [ W 0 1,2 ( Ω , ω ) ] * , where T [ W 0 1,2 ( Ω , ω ) ] * , B : W 0 1,2 ( Ω , ω ) [ W 0 1,2 ( Ω , ω ) ] * is linear and continuous, and N : W 0 1,2 ( Ω , ω ) [ W 0 1,2 ( Ω , ω ) ] * is demicontinuous and bounded and satisfies few more conditions. Further, we put Proposition 4 to this operator equation. The realization of this idea is split into 5 steps for convenience.

Step  1. We note that the operator B is linear. It follows from (17) and (20) that the operators B , N are bounded.

Step  2. Let u k u in W 0 1,2 ( Ω , ω ) . We now claim (27) N u k N u in    W 0 1,2 ( Ω , ω )    as    k or (28) | ( N u k - N u ϕ ) |    0 , i ϕ W 0 1,2 ( Ω , ω )    as    k . If u k u in W 0 1,2 ( Ω , ω ) , then | u k | | u | in L 2 ( Ω , ω ) and u k u in L 2 ( Ω , ω ) . Using Proposition 2, there exist a subsequence { u n k } and functions v 1 and v 2 in L 2 ( Ω , ω ) such that (29) u k ( x ) u ( x ) ω -a . e . in    Ω , | u n k ( x ) | v 1 ( x ) ; ω -a . e . in    Ω , u k ( x ) u ( x ) ω -a . e . in    Ω , | u n k ( x ) | v 2 ( x ) ; ω -a . e . in    Ω . Now (30) | ( N u k - N u ϕ ) | = | Ω h ( u k ( x ) , u k ( x ) ) hhhhhhhhhhhhhhh - h ( u ( x ) , u ( x ) ) g 2 ( x ) ϕ k ( x ) Ω | . Since h is bounded, we get (31) | h ( u k ( x ) , u k ( x ) ) - h ( u ( x ) , u ( x ) ) g 2 ( x ) ϕ ( x ) | 2 A | g 2 ϕ | . Also, (32) Ω | g 2 ϕ | g 2 ω 2 , Ω ϕ 2 , Ω < . By ( H 1 ) and (29), we infer that (33) [ h ( u k ( x ) , u k ( x ) ) - h ( u ( x ) , u ( x ) ) ] g 2 ( x ) ϕ ( x ) 0 hhhhhhhhhhhhhhhhhhhhhhhhh    a . e    in Ω , as k . Letting k , by dominated convergence theorem, we obtain (34) | ( N u k - N u ϕ ) | = | Ω [ h ( u k ( x ) , u k ( x ) ) - h ( u ( x ) , u ( x ) ) ] iiiiiiiii × g 2 ( x ) ϕ ( x ) Ω | 0 . Hence, we have claim (28) or equivalently N is demicontinuous.

Step  3. Next, we claim that B + N is asymptotically linear. Since h is bounded, we observe that, for all u W 0 1,2 ( Ω , ω ) , (35) | ( N u ϕ ) |    = | B 2 ( u , ϕ ) |    A C Ω    g 2 ω 2 , Ω ϕ 0,1 , 2 which implies N u    C , where C = A C Ω g 2 / ω 2 , Ω . Consequently, (36) N u    u 0,1 , 2 0 as    u 0,1 , 2 , which shows that B + N is asymptotically linear.

Step  4. We denote A t ( u ) = B u + t ( N u - T ) , t [ 0,1 ] . Let u k u in W 0 1,2 ( Ω , ω ) and (37) 0 = lim k ( B u k + t ( N u k - T ) - ( B u + t ( N u - T ) u k - u ) ) ii = lim k ( B u k - B u + t ( N u k - N u ) u k - u ) ii = lim k ( B u k - B u u k - u ) + t lim k ( N u k - N u u k - u ) . We claim that u k u strongly in W 0 1,2 ( Ω , ω ) or A t ( u ) satisfies condition ( S ) . By (2) and inequality (9) of Proposition 1, it follows that (38) B 1 ( u k - u , u k - u ) = Ω a i j D i ( u k - u ) D j ( u k - u ) d x - Ω μ ( u k - u ) 2 g 1 d x λ Ω | ( u k - u ) | 2 ω d x - Ω μ ( u k - u ) 2 g 1 d x ( λ - μ C Ω g 1 ω ) Ω | ( u k - u ) | 2 ω d x = β    Ω u k - u 0,1 , 2 2 , where β Ω = ( λ - μ C Ω g 1 / ω ) , a positive constant depending on Ω . Since B is linear, from (38) we have (39) ( B u k - B u u k - u ) = ( B ( u k - u ) u k - u ) iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii = B 1 ( u k - u , u k - u ) iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii    β    Ω u k - u 0,1 , 2 2 . By hypothesis ( H 3 ) , we have now (40) ( N u n - N u u n - u ) = Ω [ g 2 h ( u k , u k ) - h ( u , u ) ] ( u n - u ) 0 . From (37), (39), and (40), we note (41) 0 β Ω lim k u k - u 0,1 , 2 2 = 0 . Since, by condition (25), β Ω > 0 , we have (42) u k - u 0,1 , 2 2 0 , as    k . Consequently, u k - u 0,1 , 2 0 , as k , which implies that, for each t [ 0,1 ] , A t ( u ) satisfies condition (S).

Step  5. Since, by given hypothesis, μ > 0 is not an eigenvalue of (24), B u = 0 implies u = 0 . By Proposition 4, B u + N u = T has a solution u W 0 1,2 ( Ω , ω ) which equivalently shows that the BVP (3) has a solution u W 0 1,2 ( Ω , ω ) .

Remark 7.

In the following results, we dispense with condition (25), when g 1 does not change sign. The two results are related to the cases when g 1 > 0 with μ < 0 and g 1 < 0 with μ > 0 .

The proof of the following results is similar to Theorem 6 and, hence, we give a sketch of the proof.

Theorem 8.

Suppose that ( H 1 ) ( H 3 ) hold. Let g 1 > 0 and μ < 0 ; then the BVP (3) has a solution u W 0 1,2 ( Ω , ω ) .

Proof.

As in Theorem 6, the basic idea is to reduce the problem (3) to an operator equation B u + N u = T . We note that the operator B is linear and continuous and N is bounded. Let u k u in W 0 1,2 ( Ω , ω ) and as in (37) (43) 0 = lim k ( A t ( u k ) - A t ( u ) u k - u ) 1 = lim k ( B u k - B u u k - u ) + t lim k ( ( N u k - N u ) u k - u ) . Since μ < 0 and g 1 > 0 , then, by (2) and the weighted Sobolev inequality (9), it follows that (44)    B 1 ( u k - u , u k - u ) = Ω a i j D i ( u k - u ) D j ( u k - u ) d x - Ω μ ( u k - u ) 2 g 1 d x λ Ω | ( u k - u ) | 2 ω d x - Ω μ ( u k - u ) 2 g 1 d x    λ Ω | ( u k - u ) | 2 ω d x =    λ    u k - u 0,1 , 2 2 . Since B is linear, from (44) we have (45) ( B u k - B u u k - u ) iii = ( B ( u k - u ) u k - u ) = B 1 ( u k - u , u k - u ) λ u k - u 0,1 , 2 2 . By hypothesis ( H 3 ) , we have now (46) ( N u n - N u u n - u ) = Ω [ g 2 ( x ) h ( u k ( x ) , u k ( x ) ) iiiii - h ( u ( x ) , u ( x ) ) g 2 ] ( u n - u ) 0 . From (43), (45), and (46), we note that (47) 0 λ    lim k u k - u 0,1 , 2 2 = 0 . Since λ    > 0 , we have u k - u 0,1 , 2 2 0 and hence u k - u 0,1 , 2 0 , as k , which implies that, for each t [ 0,1 ] , A t ( u ) satisfies condition (S). Also, we note that B + N is asymptotically linear. Now B u = 0 implies (48) Ω a i j | D i u ( x ) | 2 d x - Ω μ u 2 ( x ) g 1 ( x ) d x = 0 or (49) λ Ω | D u ( x ) | 2    d x Ω a i j | D i u ( x ) | 2 d x ihhiiiiiiiiiiiiiiiiiiii = μ Ω u 2 ( x ) g 1 ( x ) d x which shows that u = 0 (since μ < 0 and g 1 > 0 ). By Proposition 4, B u + N u = T has a solution u W 0 1,2 ( Ω , ω ) .

With suitable changes in the proof of Theorem 8, we arrive at the following result.

Theorem 9.

Let the hypotheses of Theorem 8 hold, except that g 1 < 0 and μ > 0 . Then, the BVP (3) has a solution u W 0 1,2 ( Ω , ω ) .

Proof.

The proof of this result is in the same lines to that of Theorem 8 and hence omitted.

Remark 10.

In Theorems 8 and 9, we have studied the BVP (3) both positive and negative values of μ . In Theorem 9, with positive value of μ we do not need the extra condition (25) at the cost of g 1 negative.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Zeidler E. Nonlinear Functional Analysis and Its Applications—Part II/B 1990 New York, NY, USA Springer 10.1007/978-1-4612-0985-0 MR1033498 Cavalheiro A. C. Existence of solutions in weighted Sobolev spaces for some degenerate semilinear elliptic equations Applied Mathematics Letters 2004 17 4 387 391 10.1016/S0893-9659(04)90079-1 MR2045742 ZBL1133.35351 Franchi B. Serapioni R. Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach Annali della Scuola Normale Superiore di Pisa 1987 14 4 527 568 MR963489 ZBL0685.35046 Fabes E. Kenig C. Serapioni R. The local regularity of solutions of degenerate elliptic equations Communications in Partial Differential Equations 1982 7 1 77 116 10.1080/03605308208820218 MR643158 ZBL0498.35042 Fabes E. Jerison D. Kenig C. The Wiener test for degenerate elliptic equations Annales de l'Institut Fourier 1982 32 3 151 182 MR688024 Chanillo S. Wheeden R. L. Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions American Journal of Mathematics 1985 107 5 1191 1226 10.2307/2374351 MR805809 ZBL0575.42026 Piat V. C. Cassano F. S. Relaxation of degenerate variational integrals Nonlinear Analysis: Theory, Methods & Applications 1994 22 4 409 424 10.1016/0362-546X(94)90165-1 MR1266369 ZBL0799.49012 Raghavendra V. Kar R. Existence of weak solutions for degenerate semilinear elliptic equations in unbounded domains Electronic Journal of Differential Equations 2009 160 1 13 MR2578783 ZBL1189.35132 Turesson B. O. Nonlinear Potential Theory and Weighted Sobolev Spaces 2000 1736 Berlin, Germany Springer Lecture Notes in Mathematics 10.1007/BFb0103908 MR1774162 García-Cuerva J. de Francia J. L. R. Weighted Norm Inequalities and Related Topics 1985 116 Amsterdam, The Netherlands North-Holland North-Holland Mathematics Studies MR807149 Heinonen J. Kilpeläinen T. Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations 1993 Oxford, UK Clarendon Press Oxford Mathematical Monographs MR1207810 Kufner A. John O. Fučik S. Functions Spaces 1977 Leyden, The Netherlands Noordhoff Hess P. On the Fredholm alternative for nonlinear functional equations in Banach spaces Proceedings of the American Mathematical Society 1972 33 55 61 MR0301585 10.1090/S0002-9939-1972-0301585-9 ZBL0249.47064