A Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s

Transcription

1 A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of Craiova, Craiova, Romania b Department of Mathematics, Central European University, 1051 Budapest, Hungary c Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, Bucharest, Romania addresses: mmihailes@yahoo.com vicentiu.radulescu@imar.ro denisa.stancu@yahoo.com Abstract. Given R N (N 2) a bounded smooth domain we establish a Caffarelli-Kohn-Nirenberg type ineuality on in the case when a variable exponent, of class C 1, is involved. Our main result is proved under the assumption that there exists a smooth vector function a : R N, satisfying div a (x) > 0 and a (x) = 0 for any x. Particularly, we supplement a result of X. Fan, Q. Zhang and D. Zhao, from 2005, regarding the positivity of the first eigenvalue of the -Laplace operator. Moreover, we provide an application of our result to the study of degenerate PDE s involving variable exponent growth conditions Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 35P05. Key words: Caffarelli-Kohn-Nirenberg type ineuality, eigenvalue problem, degenerate elliptic euation, variable exponent, critical point. 1 Introduction In 1984, L. Caffarelli, R. Kohn and L. Nirenberg proved in [3], in the context of some more general ineualities, the following result: given p (1, N), for all u C 1 c (), there exists a positive constant C a,b such that where ( p/ x b u dx) C a,b < a < N p, a b a + 1, = p x ap u p dx, (1.1) Np N p(1 + a b), Correspondence address: Mihai Mihăilescu, Department of Mathematics, University of Craiova, Craiova, Romania. mmihailes@yahoo.com 1

2 and R N is an arbitrary open domain. Note that the Caffarelli-Kohn-Nirenberg ineuality (1.1) reduces to the classical Sobolev ineuality (if a = b = 0) and to the Hardy ineuality (if a = 0 and b = 1). Ineuality (1.1) proves to be an important tool in studying degenerate elliptic problems. It is also related with the understanding of some important phenomena such as best constants, existence or nonexistence of extremal functions, symmetry properties of minimizers, compactness of minimizing seuences, concentration phenomena, etc. applications of the Caffarelli-Kohn-Nirenberg ineuality. We refer to [4, 5, 6, 1, 8, 9, 13, 16, 19, 21] for relevant In the years that followed this ineuality was extensively studied (see, e.g. [4, 5, 6, 21] and the references therein). An important conseuence of the Caffarelli-Kohn-Nirenberg ineuality is that it enabled the study of some degenerate elliptic euations which involve differential operators of the type div(a(x) u(x) p ), where a(x) is a nonnegative function satisfying inf x a(x) = 0. Thus, the resulting operator is not uniformly elliptic and conseuently some of the techniues that can be applied in solving euations involving uniformly elliptic operators fail in this new context. Degenerate differential operators involving a non negative weight that is allowed to have zeros at some points or even to be unbounded are used in the study of many physical phenomena related to euilibrium of anisotropic continuous media. The goal of this paper is to obtain ineualities of type (1.1) in the case when the constant p is replaced by a function of class C 1 and to use them in studying some degenerate elliptic euations involving variable exponent growth conditions. Our attempt will be considered in the context of bounded smooth domains R N with N 2. Particularly, we supplement the result of X. Fan, Q. Zhang and D. Zhao [7, Theorem 3.3], regarding the positivity of the first eigenvalue of the -Laplace operator. 2 Variable exponent Lebesgue and Sobolev spaces We recall some definitions and basic properties of the Lebesgue Sobolev spaces L () and W 1, 0 (), where is a bounded domain in R N and : (1, ) is a continuous function. For further information and proofs we refer to Kováčik and Rákosník [12] and Musielak [15]. On the other hand, regarding applications of variable exponent Lebesgue and Sobolev spaces to PDE s we refer to Harjulehto, Hästö, Lê and Nuortio [11] while for some physical motivations of such problems we remember the contributions of Rajagopal and Ruzicka [17], Ruzicka [18] and Zhikov [23]. For any continuous function h : (1, ) we define h + = sup h(x) and h = inf h(x). x x Given C(, (1, )), we define the variable exponent Lebesgue space L () = {u; u is a measurable real-valued function such that u(x) dx < }. 2

3 L () endowed with the Luxemburg norm, that is { u = inf µ > 0; u(x) µ dx 1 }, is a reflexive Banach space. If p 1, p 2 are variable exponents so that p 1 (x) p 2 (x) almost everywhere in then there exists the continuous embedding L p 2(x) () L p 1(x) (), whose norm does not exceed + 1. We denote by L (x) () the conjugate space of L (), where 1/ + 1/(x) = 1. For any u L () and v L (x) () the Hölder type ineuality ( uv dx 1 p + 1 ) u v (x) (2.1) holds true. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L () space, which is the mapping ρ : L () R defined by ρ (u) = u dx. If (u n ), u L () and p + < then the following relations hold true Next, we define W 1, 0 () as the closure of C 0 u > 1 u p ρ (u) u p+ (2.2) u < 1 u p+ ρ (u) u p (2.3) u n u 0 ρ (u n u) 0 (2.4) u n ρ (u n ). (2.5) u = u. () under the norm The space (W 1, 0 (), ) is a separable and reflexive Banach space. We note that if C + () and (x) < p (x) for all x then the embedding W 1, 0 () L (x) () is compact and continuous, where p (x) = N N if < N or p (x) = + if N. 3 A Caffarelli-Kohn-Nirenberg type ineuality in bounded domains involving variable exponent growth conditions Assume R N (N 2) is an open, bounded and smooth set. For each x, x = (x 1,..., x N ) and i {1,..., N} we denote m i = inf x x i M i = sup x i. x 3

4 For each i {1,..., N} let a i : [m i, M i ] R be functions of class C 1. Particularly, the functions a i are allowed to vanish. Let a : R N be defined by a (x) = (a1 (x 1 ),..., a N (x N )). We assume that there exists a 0 > 0 a constant such that div a (x) a 0 > 0, x. (3.1) Next, we consider p : (1, N) is a function of class C 1 satisfying a (x) = 0, x. (3.2) We prove the following result: Theorem 1. Assume that a (x) and are defined as above and satisfy conditions (3.1) and (3.2). Then there exists a positive constant C such that u(x) dx C a (x) u(x) dx, u Cc 1 (). (3.3) Proof. The proof of Theorem 1 is inspired by the ideas in [22, Théorème 20.7]. Simple computations based on relation (3.2) show that for each u C 1 c () the following euality holds true div( u(x) a (x)) = N i=1 ( ) u(x) a i (x i ) x i = u(x) div a (x) N [ + a i (x i ) u(x) 2 u(x) u + u(x) log( u(x) ) p ] x i x i i=1 = u(x) div a (x) + u(x) 2 u(x) u(x) a (x) + u(x) log( u(x) ) a (x) = u(x) div a (x) + u(x) 2 u(x) u(x) a (x) On the other hand, the flux-divergence theorem implies that for each u Cc 1 () we have div( u(x) a (x)) dx = u(x) a (x) n dσ(x) = 0. Using the above pieces of information we infer that for each u Cc 1 () it holds true u(x) div a (x) dx p + u(x) 1 u(x) a (x) dx. 4

5 Next, we recall that for each ɛ > 0, each x and each A, B 0 the following Young type ineuality holds true (see, e.g. [2, the footnote on p. 56]) We fix ɛ > 0 such that AB ɛa ɛ 1 B. p + ɛ < a 0, where a 0 is given by relation (3.1). The above facts and relation (3.1) yield a 0 for any u C 1 c (), or u(x) dx p [ɛ + u(x) dx + (a 0 ɛp + ) u(x) dx [ (1 ɛ ) p 1 + ( ) 1 1 a (x) u(x) dx], ɛ ( ) ] 1 p + 1 p + a (x) u(x) dx, ɛ for any u C 1 c (). The conclusion of Theorem 1 is now clear. Remark 1. The result of Theorem 1 implies the fact that under the hypotheses (3.1) and (3.2) there exists a positive constant D such that u(x) dx D u(x) dx, u C 1 c (). Thus, we deduce that in the hypotheses of Theorem 1 we have u dx inf > 0. (3.4) u W 1, 0 ()\{0} u dx The above relation asserts that in this case the first eigenvalue of the -Laplace operator (i.e., u := div( u 2 u)) is positive. That fact is not obvious as X. Fan, Q. Zhang and D. Zhao pointed out in [7]. Actually, the infimum of the set of eigenvalues corresponding to the -Laplace operator can be 0 (see, [7, Theorem 3.1]). On the other hand, a necessary and sufficient condition such that (3.4) holds true has not been obtained yet excepting the case when N = 1 (in that case, the infimum is positive if and only if is a monotone function, see [7, Theorem 3.2]). However, the authors of [7] pointed out that in the case N > 1 a sufficient condition to have (3.4) is to exist a vector l R N \ {0} such that, for any x, the function f(t) = p(x + tl) is monotone, for t I x := {s; x + sl } (see [7, Theorem 3.3]). Assuming p is of class C 1 the monotony of function f reads as follows: either p(x + tl) l 0, for all t I x, x, 5

6 or p(x + tl) l 0, for all t I x, x. The above conditions seem to be related to condition (3.2) in our paper. On the other hand, in the case when N = 1 relation (3.2) implies that should be a constant function. The two results do not contradict each other but they seem to supplement each other. Example 1. We point out an example of functions a (x) and satisfying conditions (3.1) and (3.2) in the case when a (x) can vanish in some points of. Let N 3 and = B 1 N (0), the ball centered in the origin of radius 1 N. We define a (x) : R N by a (x) = ( x1, x 2, x 3,..., x N 1, x N ), (more exactly, function a (x) is associated to a vector x the vector obtained from x by changing in the first position x 1 by x 1 and keeping unchanged x i for i {2,..., N}). Clearly, a (x) is of class C 1, a (0) = 0 and we have Thus, condition (3.1) is satisfied. Next, we define p : (1, N) by div( a (x)) = N 2 1, x. = x 1 (x 2 + x x N 1 + x N ) + 2, x. It is easy to check that p is of class C 1 and some elementary computations show that a (x) = (x x N )( x 1 ) + x 1 x x 1 x N = 0, x. It means that condition (3.2) is satisfied, too. Example 2. We point out a second example, for N = 2. Taking = B 1 (0), a (x) = ( x 1, 2x 2 ) 3 1/3 and = x 2 1 x it is easy to check that relations (3.1) and (3.2) are fulfilled. Remark 2. If N, a and p are as in Example 1 or Example 2 then the result of Theorem 1 reads as follows: there exists a positive constant C > 0 such that u(x) dx C x u(x) dx, u Cc 1 (). (3.5) 4 Applications in solving PDE s involving variable exponent growth conditions In this section we assume that N,, a (x) and are as in Example 1 or Example 2. We denote by D 1, 0 () the closure of C 1 c () under the norm u = x u(x). Undoubtedly, (D 1, 0 (), ) is a reflexive Banach space. 6

7 4.1 A compact embedding result We prove the following result: Theorem 2. Assume that N,, a (x) and are as in Example 1 or Example 2 and p > ( Then D 1, 0 () is compactly embedded in L () for each 1, 2Np 2N+p ). 2N 2N 1. Proof. Let {u n } be a bounded seuence in D 1, 0 (). There exists ɛ 0 (0, 1) such that we have B ɛ0 (0). Let ɛ (0, ɛ 0 ) be arbitrary but fixed. It is obvious that {u n } D 1, 0 ( \ B ɛ (0)) is a bounded seuence. The classical compact embedding theorem for variable exponent spaces shows that there exists a convergent subseuence of {u n }, still denoted by {u n }, in L ( \ B ɛ (0)). Thus, for any n and m large enough we have \B ɛ (0) u n u m dx < ɛ. (4.1) On the other hand, the Hölder ineuality for variable exponent spaces implies u n u m dx = x 2 x 2 un u m dx B ɛ (0) where D 1 is a positive constant. B ɛ (0) x D 1 2 χbɛ (0) ( Furthermore, ineuality (3.5) and relations (2.2) and (2.3) imply x 2 un u m where D 2 is a positive constant. ( ) x 2 un u m dx [ (sup x + 1) p + 2p x p + + (sup x + 1) 2 x [ ( D 2 x (u n u m ) dx ( ) x 2 un u m x ) 2 un u m p dx + ] [ ρ (u n u m ) p ) p + (, + ρ (u n u m ) p + ] x (u n u m ) dx Combining the above pieces of information we find that there exists a positive constant M such that u n u m dx M ). B ɛ(0) x 2 χbɛ(0) But using again relations (2.2) and (2.3) it is easy to see that x 2 χbɛ (0) ( ) ρ( ) ( x 2 χbɛ (0) ) ( ( ) ) + + ρ( ( ) ( x 2 χbɛ (0) ) ( ( ) ), ) ] p + 7

8 where ( ) =, and assuming ɛ (0, 1) B ɛ (0) x 2( ) dx = B ɛ (0) ɛ 0 = ω N 1 α ɛα, x p 2(p ) dx ω N r N 1 r p 2(p ) dr where α = N p 2(p ) > 0 and ω N is the area of the unit ball in R N. Conseuently, u n u m dx M 1 (ɛ α 1 + ɛ α 2 ), B ɛ (0) with α 1, α 2 > 0 and M 1 > 0 a constant. The above ineuality and relation (4.1) show that for any n and m large enough we have u n u m dx M 2 (ɛ + ɛ α 1 + ɛ α 2 ), where M 2 is a positive constant. We infer that {u n } is a Cauchy seuence in L () and conseuently D 1, 0 () is compactly embedded in L (). The proof of Theorem 2 is complete. Remark 3. The proof of Theorem 2 still holds true if we replace the space L () by L (x) (), where : (1, ) is a continuous function satisfying 1 < + < 2Np 2N+p. 4.2 Existence of solutions for a singular PDE involving variable exponent growth conditions Assume (x) is a function satisfying the hypotheses given in Remark 3. We investigate the existence of solutions of the problem div( x u(x) 2 u(x)) = λ u(x) (x) 2 u(x) for x, u(x) = 0 for x, (4.2) where λ is a positive constant. We say that u D 1, 0 () is a weak solution of problem (4.2) if x u 2 u v dx λ u (x) 2 uv dx = 0, v D 1, 0 (). We show the following existence result on problem (4.2): Theorem 3. For each λ > 0 problem (4.2) has a nontrivial weak solution. 8

9 Proof of Theorem 3. In order to prove Theorem 3 we define, for each λ > 0, the energetic functional associated to problem (4.2), J λ : D 1, 0 () R by J λ (u) = x u dx λ 1 (x) u (x) dx, for each u D 1, 0 (). Standard arguments (see, e.g. [14]) show that J λ C 1 (D 1, 0 (), R) and its derivative is given by J λ (u), v = x u 2 u v dx λ u (x) 2 uv dx, for all u, v D 1, 0 (). We infer that u is a solution of problem (4.2) if and only if it is a critical point of J λ. Conseuently, we concentrate our efforts on finding critical points for J λ. In this context we prove the following assertions: (a) The functional J λ is weakly lower semi-continuous; (b) The functional J λ is bounded from below and coercive; (c) There exists ψ D 1, 0 () \ {0} such that J λ (ψ) < 0. The arguments to prove (a), (b) and (c) are detailed below. (a) Similar arguments as in the proof of [14, Proposition 3.6 (ii)] can be used in order to obtain the fact that J λ is weakly lower semi-continuous. (b) It is obvious that for any u D 1, 0 () we have J λ (u) 1 p + x u dx λ u (x) dx 1 p + x u dx λ ( u (x) + u + (x) ). If u > 1 the above ineuality and Theorem 2 imply that there exists a positive constant K such that J λ (u) 1 p + u p Kλ ( u + u + ). Taking into account that 1 < + < 2Np 2N+p < p the above ineuality shows that lim u J λ (u) =, that is J λ is coercive. On the other hand, it is clear that for any u D 1, 0 () we have J λ (u) 1 p + min{ u p, u p+ } Kλ ( u + u + ), and thus, we deduce that J λ is bounded from below. (c) We fix ϕ Cc 1 (), ϕ 0. Then for each t (0, 1) we have x t J λ (tϕ) = ϕ t (x) dx λ (x) ϕ (x) dx x 1 t p ϕ dx λt + (x) ϕ (x) dx. 9

10 Thus, there exist L 1 and L 2 two positive constants such that for each t (0, 1) we have J λ (tϕ) L 1 t p L 2 t +. Taking into account that + < p, by the above ineuality we infer that for any t (0, min{1, ( L 2 L 1 ) 1/(p +) }) we have J λ (tϕ) < 0. Next, we deduce by (a) and (b) that J λ is weakly lower semi-continuous, bounded from below and coercive. These facts in relation with [20, Theorem 1.2 ] show that there exists u λ D 1, 0 () a global minimum point of J λ. Moreover, since (c) holds true it follows that u λ 0. Standard arguments based on Theorem 2 show that u λ is actually a critical point of J λ and thus, a nontrivial weak solution of problem (4.2). The proof of Theorem 3 is complete. Acknowledgments. The first two authors have been supported by Grant CNCSIS PNII 79/2007 Degenerate and Singular Nonlinear Processes. References [1] M. Badiale and G. Tarantello, A Sobolev-Hardy ineuality with applications to a nonlinear elliptic euation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), [2] H. Brezis, Analyse Fonctionnelle. Théorie, Méthodes et Applications, Masson, Paris, [3] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation ineualities with weights, Compositio Math. 53 (1984), [4] F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg ineualities: sharp constants, existence (and nonexistence) and symmetry of extremal function, Commun. Pure Appl. Math. 54 (2001), [5] P. Caldiroli and R. Musina, On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calc. Var. Partial Differential Euations 8 (1999), [6] P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Euations Appl. 7 (2000), [7] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of -Laplacian Dirichlet problem, Journal of Mathematical Analysis and Applications 302 (2005), [8] M. Ghergu and V. Rădulescu, Singular elliptic problems with lack of compactness, Ann. Mat. Pura Appl. 185 (2006), [9] N. Ghoussoub and C. Yuan, Multiple solutions for uasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Euations of Second Order, Springer, Berlin, [11] P. Harjulehto, P. Hästö, Ú. V. Lê and M. Nuortio, Overview of differential euations with non-standard growth, preprint. [12] O. Kováčik and J. Rákosník, On spaces L and W 1,, Czechoslovak Math. J. 41 (1991),

11 [13] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related ineualities, Ann. Math. 118 (1983), [14] M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. Roy. Soc. London Ser. A 462 (2006), [15] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin, [16] P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for p-laplacain euations with singular weights, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), [17] K. R. Rajagopal and M. Ruzicka, Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn. 13 (2001), [18] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Verlag, Berlin, [19] D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Euations 18 (2003), [20] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Euations and Hamiltonian Systems, Springer, Heidelberg, [21] Z.-Q. Wang and M. Willem, Singular minimization problems, J. Differential Euations 161 (2000), [22] M. Willem, Analyse Fonctionnelle Elémentaire, Cassini, [23] V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987),