Journal of Alied Analysis and Comutation Volume 7, Number 2, May 2017, 659 669 Website:htt://jaac-online.com/ DOI:10.11948/2017041 KIRCHHOFF TYPE PROBLEMS INVOLVING P -BIHARMONIC OPERATORS AND CRITICAL EXPONENTS Nguyen Thanh Chung and Pham Hong Minh Abstract In this aer, we study the existence of solutions for a class of Kirchhoff tye roblems involving -biharmonic oerators and critical exonents. The roof is essentially based on the mountain ass theorem due to Ambrosetti and Rabinowitz [2] and the Concentration Comactness Princile due to Lions [18, 19]. Keywords Kirchhoff tye roblems, -biharmonic oerators, critical exonents, mountain ass theorem. MSC(2010) 47A75, 35B38, 35P30, 34L05, 34L30. 1. Introduction and Preliminaries In this aer, we are interested in the existence of solutions for the following Kirchhoff tye roblem { ( M u dx ) ( ) u 2 u = λf(x, u) + u 2 u, x, u = u ν = 0, x, (1.1) where is a bounded domain in R N with C 2 boundary, N 2, is the Lalace oerator and u ν is the outer normal derivative, λ is a ositive arameter, f : R R is a Carathéodory function, is the critical exonent, i.e. = { N N 2, if 2 < N, +, if 2 N, (1.2) where (2, + ). Throughout this aer we assume that M : [0, + ) R is a continuous and increasing function that satisfies: (M 0 ) There exists m 0 > 0 such that M(t) m 0 = M(0) for all t [0, + ). Since roblem (1.1) contains integral over, it is no longer a ointwise identity; therefore it is often called nonlocal roblem. This roblem models several hysical and biological systems, where u describes a rocess which deends on the average of itself, such as the oulation density, see [10, 17]. Kirchhoff tye roblems for the corresonding author. Email address: ntchung82@yahoo.com (N.T. Chung) Deartment of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam The work is suorted by Vietnam National Foundation for Science and Technology Develoment (NAFOSTED).
660 N.T. Chung & P.H. Minh -Lalace oerators have been studied by many mathematicians, see [4, 5, 9, 13, 14, 21, 22, 24]. In a recent aer [11], Colasuonno and Pucci have introduced - olyharmonic oerators. Using variational methods, they studied the multilicity of solutions for a class of (x)-olyharmonic ellitic Kirchhoff equations. In [20], V.F. Lubyshev studied the existence of solutions for an even-order nonlinear roblem with convex-concave nonlinearity. In [6], the author extended the revious results in [11] to nonlocal higher-order roblems. Althought the Kirchhoff function M(t) in [6, 11] was assumed to be degenerate at zero, the nonlinearity is not critical. Finally, we refer the readers to two related interesting aers [7, 25]. In [7], G. Autuori et al. considered a class of Kirchhoff tye roblems involving a fractional ellitic oerator and a critical nonlinearity while in [25], L. Zhao et al. studied the existence of solutions for a higher order Kirchhoff tye roblem with exonential critical growth. In this aer, we study the existence of solutions for Kirchhoff tye roblems involving -biharmonic oerators and critical exonents. We are motivated by the results introduced in [1, 6, 11] and some aers on -biharmonic oerators with critical exonents [3,8,12,15,16,23]. We also get a riori estimates of the obtained solution. We believe that this is the first contribution to the study of the existence of solutions for Kirchhoff tye roblems involving -biharmonic oerators with critical exonents. Due to the resence of the critical exonents, the roblem considered here is lack of comactness. To overcome this difficulty, we use the Concentration Comactness Princile due to Lions [18, 19]. The existence of a nontrivial solution is then obtained by the mountain ass theorem due to Ambrosetti and Rabinowitz in [2]. In order to state the main result, we assume that f : R R is a Carathéodory function satisfying the following conditions: (F 1 ) f(x, t) C(1 + t q 1 ) for all (x, t) R, where < q < and is defined by (1.2); (F 2 ) lim t 0 f(x,t) t 1 = 0 uniformly for x ; (F 3 ) There exists θ (, ) such that 0 < θf (x, t) f(x, t)t, x, t R\{0}, where F (x, t) = t f(x, s) ds. 0 Let W 2, 0 () be the usual Sobolev sace with resect to the norm ( ) 1 u = u dx, u W 2, 0 (). We then have that W 2, 0 () is continuously and comactly embedded into the Lebesgue sace L r () endowed the norm u r = ( u r dx ) 1 r, 2 < r <. Denote by S r the best constant for this embedding, that is, S r u r u for all u W 2, 0 (). Definition 1.1. We say that u W 2, 0 () is a weak solution of roblem (1.1) if M ( u ) u 2 u v dx λ f(x, u)v dx u 2 uv dx = 0 for all v W 2, 0 ().
Kirchhoff tye roblems involving -biharmonic oerators 661 The main result of this aer can be stated as follows. Theorem 1.1. Assume that (M 0 ), (F 1 )-(F 3 ) are satisfied. Then, there exists λ > 0 such that for all λ λ, roblem (1.1) has a nontrivial solution. Moreover, if u λ is a solution of roblem (1.1) then lim λ + u λ = 0. 2. Proof of the main result Here we are assuming, without loss of generality, that the Kirchhoff function M(t) is unbounded. Contrary case, the truncation on M(t) is not necessary. Since we are intending to work with N 2, we shall make a truncation on M as follows. From (M 0 ), given a R such that m 0 < a < θ m 0, there exists t 0 > 0 such that M(t 0 ) = a. We set { M(t), 0 t t 0, M a (t) := (2.1) a, t t 0. From (M 0 ) we get M a (t) a, t 0. (2.2) As we shall see, the roof of Theorem 1.1 is based on a careful study of the solutions of the following auxiliary roblem ( ) M a ( u ) u 2 u = λf(x, u) + u 2 u, x, (2.3) u = u ν = 0, x, where f, N,, λ are as in Section 1. We shall rove the following auxiliary result. Theorem 2.1. Assume that (M 0 ), (F 1 )-(F 3 ) are satisfied. Then, there exists λ 0 > 0 such that for all λ λ 0 and all a (m 0, θ m 0), roblem (2.3) has a nontrivial solution. We recall that u W 2, 0 () is a weak solution of roblem (2.3) if M a ( u ) u 2 u v dx λ f(x, u)v dx u 2 uv dx = 0, for all v W 2, 0 (). Hence, we shall look for nontrivial solutions of (2.3) by finding critical oints of the C 1 functional I a,λ : W 2, 0 () R given by the formula I a,λ (u) = 1 M a ( u ) λ F (x, u) dx 1 u dx, where M(t) = t 0 M(s) ds and F (x, t) = t f(x, s) ds. Note that 0 I a,λ(u)(v) = M a ( u ) u 2 u v dx λ f(x, u)v dx u 2 uv dx for all u, v W 2, 0 (). We say that a sequence {u n } W 2, 0 () is a Palais-Smale sequence for the functional I a,λ at level c R if I a,λ (u n ) c and I a,λ(u n ) 0 in (W 2, 0 ()),
662 N.T. Chung & P.H. Minh where (W 2, 0 ()) is the dual sace of W 2, 0 (). If every Palais-Smale sequence of I a,λ has a strong convergent subsequence, then one says that I a,λ satisfies the Palais-Smale condition ((P S) condition for short). Lemma 2.1. For all λ > 0, there exist ositive constants ρ and r such that I a,λ (u) r > 0 for all u W 2, 0 () with u = ρ. Proof. which imlies that From (F 1 ) and (F 2 ), for any ɛ > 0, there is C ɛ > 0 such that f(x, t) ɛ t 1 + C ɛ t q 1, (x, t) R, F (x, t) ɛ t + C ɛ q t q, (x, t) R. (2.4) From (2.4) and (M 0 ), for all u W 2, 0 (), we get I a,λ (u) = 1 M a ( u ) λ F (x, u) dx 1 m ( 0 ɛ u λ u + C ) ɛ q u q u dx u dx dx 1 m 0 u λɛs u λc ɛ Sq q u q inf u W 2, 0 ()\{0} 1 S u, where S is the best constant of the embedding W 2, 0 () L (), that is, S = u dx dx ( ). (2.5) u For λ > 0, let ɛ = m0s 2λ, we get I a,λ (u) m 0 2 u λc ɛ Sq q u q 1 ( = u m 0 2 λc ɛsq q u q S u 1 S u Since < q <, there exist ositive constants ρ and r such that I a,λ (u) r > 0 for all u W 2, 0 () with u = ρ. Lemma 2.2. For all λ > 0, there exists e W 2, 0 () with I a,λ (e) < 0 and e > ρ. Proof. From (F 3 ), there are ositive constans C 1 and C 2 such that F (x, t) C 1 t θ C 2, (x, t) R. (2.6) Fix u 0 C0 ()\{0} with u 0 = 1. Using (2.2) and (2.6), for all t > 0 large enough, we have I a,λ (tu 0 ) = 1 M a ( tu 0 ) λ F (x, tu 0 ) dx 1 tu 0 dx a t λc 1 t θ u 0 θ dx + λc 2 t ) u 0 dx..
Kirchhoff tye roblems involving -biharmonic oerators 663 Since θ (, ), the result follows by considering e = t u 0 for some t > 0 large enough. Using a version of the Mountain ass theorem due to Ambrosetti and Rabinowitz [2], without (P S) condition, there exists a sequence {u n } W 2, 0 () such that where and Γ = Lemma 2.3. It holds that I a,λ (u n ) c a,λ, I a,λ(u n ) 0 as n, c a,λ = inf γ Γ max t [0,1] I a,λ(γ(t)), { } γ C([0, 1], W 2, 0 ()) : γ(0) = 0, I a,λ (γ(1)) < 0. lim c a,λ = 0. λ + Proof. Since the functional I a,λ has the mountain ass geometry, it follows that there exists t λ > 0 verifying I a,λ (t λ u 0 ) = max t0 I a,λ (tu 0 ), where u 0 is the function d given by Lemma 2.2. Hence, dt I a,λ(t λ u 0 )(t λ u 0 ) = 0 or 0 = M a ( t λ u 0 ) t λ u 0 dx λ f(x, t λ u 0 )t λ u 0 dx t λ u 0 dx. Hence, t λ M a(t λ ) = λ From (2.2), (2.7) and (F 3 ), we get f(x, t λ u 0 )t λ u 0 dx + t λ a t λ u 0 dx, u 0 dx. (2.7) which imlies that {t λ } is bounded. Thus, there exist a sequence λ n + and t 0 such that t λn t as n. Consequently, there is C 3 > 0 such that and n N, λ n t λ n M a (t λ n ) C 3, n N, f(x, t λn u 0 )t λn u 0 dx + t λ n If t > 0, by the Dominated Convergence Theorem, f(x, t λn u 0 )t λn u 0 dx = f(x, tu 0 )tu 0 dx lim and thus (2.8) leads to (λ n lim f(x, t λn u 0 )t λn u 0 dx + t λ n u 0 dx C 3. (2.8) ) u 0 dx = +, which is an absurd. Thus, we conclude that t = 0. Now, let us consider the ath γ (t) = te for t [0, 1], which belongs to Γ, to get the following estimate 0 < c a,λ max t [0,1] I a,λ(γ (t)) = I a,λ (t λ u 0 ) M a (t λ ).
664 N.T. Chung & P.H. Minh In this way, which leads to lim λ + c a,λ = 0. lim M a (t λ ) = 0, λ + Lemma 2.4. Let {u n } W 2, 0 () be a sequence such that Then {u n } is bounded. I a,λ (u n ) c a,λ, I a,λ(u n ) 0 as n. (2.9) Proof. Assuming by contradiction that {u n } is not bounded in W 2, 0 (), u to a subsequence, we may assume that u n + as n. It follows from (2.1), (M 0 ) and (F 3 ) that for n large enough 1 + c a,λ + u n I a,λ (u n ) 1 θ I a,λ(u n )(u n ) = 1 M a ( u n ) λ F (x, u n ) dx 1 u n dx 1 θ M a ( u n ) u n dx + λ f(x, u n )u n dx + 1 θ θ ( m0 a ) u n λ (f(x, u n )u n θf (x, u n )) dx + θ θ ( ) m0 u n C 4, a θ u n dx θ 1 ) u n dx where C 4 is a ositive constant. Since a < m0 θ and θ <, the sequence {u n } is bounded. Proof of Theorem 2.1. From Lemma 2.3, we have lim c a,λ = 0. (2.10) λ + Therefore, there exists λ 0 > 0 such that c a,λ < θ 1 ) S N 2, (2.11) for all λ λ 0, where S is given by (2.5). Now, fix λ λ 0 and let us show that roblem (2.1) admits a nontrivial solution. From Lemmas 2.3 and 2.4, there exists a bounded sequence {u n } W 2, 0 () verifying I a,λ (u n ) c a,λ, I a,λ(u n ) 0 as n. (2.12) Hence, u to subsequences, we may assume that {u n } converges weakly to u W 2, 0 (), {u n } converges strongly to u in L q (), 2 < q < and u n (x) u(x) for a.e. x as n. Using the Concentration Comactness Princile due to Lions [18,19], if u n µ, u n ν weakly- in the sense of measures, where µ and ν are bounded nonnegative measures on R N, then there exist an at most countable set J, sequences
Kirchhoff tye roblems involving -biharmonic oerators 665 (x j ) j J and (ν j ) j J, (µ j ) j J nonnegative numbers such that ν = u + j J ν j δ xj, ν j > 0, µ u + j J µ j δ xj, µ j > 0, (2.13) ν j µ j S, for all j J, where δ xj is the Dirac mass at x j. Now, we claim that J =. Arguing by contradiction, assume that J and fix j J. For ɛ > 0, consider φ j,ɛ C (R N ) such that φ j,ɛ 1 in B ɛ (x j ), φ j,ɛ 0 on \B 2ɛ (x j ) and φ j,ɛ 2C ɛ, and φ j,ɛ 2C ɛ, where x 2 j belongs to the suort of ν. Since {φ j,ɛ u n } is bounded in the sace W 2, 0 (), it then follows from (2.1) that I a,λ (u n)(φ j,ɛ u n ) 0 as n, that is, 2M a ( u n ) u n 2 u n ( u n φ j,ɛ )dx + M a ( u n ) u n u n 2 u n φ j,ɛ dx = M a ( u n ) u n φ j,ɛ dx + λ f(x, u n )φ j,ɛ u n dx + u n φ j,ɛ dx + o n (1). (2.14) Using the Hölder inequality and the boundedness of the sequence {u n }, we have u n 2 u n ( u n φ j,ɛ ) dx = u n 2 u n ( u n φ j,ɛ ) dx B 2ɛ(x j) u n 1 u n φ j,ɛ dx B 2ɛ(x j) ( u n dx B 2ɛ(x j) ) 1 ( u n φ j,ɛ dx B 2ɛ(x j) ) 1 ( C 3 u n φ j,ɛ dx B 2ɛ(x j) ( ) 1 ( ) 2 N C 3 u n φ j,ɛ 2N B 2ɛ(x j) B 2ɛ(x j) ( ) 1 C 3 u n 0 as n and ɛ 0. B 2ɛ(x j) ) 1 (2.15) Since {u n } is bounded, we may assume that u n t 1 0 as n. Observing that M(t) is continuous, we then have M( u n ) M(t 1 ) m 0 > 0 as n.
666 N.T. Chung & P.H. Minh Hence, M a ( u n ) Similarly, we have lim M a ( u n ) u n 2 u n ( u n φ j,ɛ ) dx 0 as n and ɛ 0. (2.16) u n u n 2 u n φ j,ɛ dx = 0. (2.17) On the other hand, by (F 2 ) and the boundedness the sequence {u n } in W 2, 0 () we also have lim f(x, u n )φ j,ɛ u n dx = 0. (2.18) From (2.14)-(2.18), letting n, we deduce that dν M a (t 1 ) φ j,ɛ dµ + o ɛ (1). Letting ɛ 0 and using the standard theory of Radon measures, we conclude that ν j M a (t 1 )µ j m 0 µ j. Using (2.13) we have ν j S N 2, (2.19) where S is given by (2.5). Now, we shall rove that (2.19) cannot occur, and therefore the set J =. Indeed, arguing by contradiction, let us suose that ν j S N 2 for some j J. Since {u n } is a (P S) ca,λ sequence for the functional I a,λ, from the conditions (F 3 ) and (M 0 ), and m 0 < a < θ m 0 we have c a,λ = I a,λ (u n ) 1 θ I a,λ(u n )(u n ) + o n (1) 1 M a ( u n ) 1 θ M a( u n ) u n + ( m0 a ) u n + θ θ 1 ) ( 1 θ 1 ) u n dx + o n (1) θ 1 ) Letting n in (2.20), we get and then θ 1 ) u n dx + o n (1) u n dx + o n (1) u n φ j,ɛ dx + o n (1). (2.20) c a,λ c a,λ θ 1 ) φ j,ɛ dν θ 1 ) S N 2, (2.21) which contradicts (2.11). Thus, J = and it follows that u n dx = u dx. (2.22) lim
Kirchhoff tye roblems involving -biharmonic oerators 667 We also have u n (x) u(x) a.e. x as n, so by the condition (F 1 ) and the Dominated Convergence Theorem, we deduce that (f(x, u n )u n f(x, u)u) dx = 0. (2.23) lim From (2.22)-(2.23) we deduce since I a,λ (u n)(u n ) 0 as n that lim M a ( u n ) u n = λ f(x, u)u dx + u dx. (2.24) On the other hand, by (2.12), and the boundedness of u n u we have I a,λ (u n)(u n u) 0 as n, that is, M a ( u n ) u n 2 u n ( u n u) dx λ f(x, u n )(u n u) dx (2.25) u n 2 u n (u n u) dx 0 as n. Using Hölder s inequality we have f(x, u n )(u n u) dx f(x, u n ) u n u dx C (1 + u n q 1 ) u n u dx C ( q 1 q 0 as n + u n q 1 L q () ) u n u L q () (2.26) and u n 2 u n (u n u) dx u n 1 u n u dx u n 1 L () u n u L () 0 as n. (2.27) From (2.25)-(2.27) we get M a ( u n ) u n 2 u n ( u n u) dx 0 as n. (2.28) Using the condition (M 0 ), it follows that u n 2 u n ( u n u) dx 0 as n. (2.29) By standard arguments, we can show that {u n } converges stronlgy to u in W 2, 0 (). This comletes the roof of Theorem 2.1. Now, we are in the osition to rove Theorem 1.1. Proof of Theorem 1.1. Let λ 0 be as in Theorem 2.1 and, for λ λ 0, let u λ W 2, 0 () be the nontrival solution of roblem (2.3) found in Theorem 2.1. We
668 N.T. Chung & P.H. Minh claim that there exists λ λ 0 such that u λ t 0 for all λ λ. If this is the case, it follows from the definition of M a (t) that M a ( u λ ) = M( u λ ). Thus, u λ is a nontrivial weak solution of roblem (1.1). We argue by contradiction that, there is a sequence {λ n } R such that λ n + as n and u λn t 0. Then we have c a,λn 1 M a ( u λn ) 1 ( θ M a( u λn ) u λn m0 a ) u λn θ ( m0 a ) (2.30) t 0, θ which is a contradiction since lim c a,λn = 0 and a (m 0, θ m 0). Finally, we shall rove that lim λ + u λ = 0. Indeed, by (M 0 ) and the fact that u λ t 0, it follows that M( u λ ) M(t 0 ) = a. Hence, using (M 0 ) and (F 4 ) we have c a,λ 1 M( u λ ) 1 θ M( u λ ) u λ m 0 u λ a θ u λ = ( m0 a θ ) u λ. (2.31) Using Lemma 2.4 again we have that lim λ + c a,λ = 0. Therefore, it follows since a (m 0, θ m 0) that lim λ + u λ = 0. The roof of Theorem 1.1 is now comleted. Acknowledgements The authors would like to thank the referees for their suggestions and helful comments which imroved the resentation of the original manuscrit. This research is suorted by Vietnam National Foundation for Science and Technology Develoment (NAFOSTED) (Grant N.101.02.2017.04). References [1] G. A. Afrouzi, M. Mirzaour and N. T. Chung, Existence and multilicity of solutions for Kirchhoff tye roblems involving (x)-biharmonic oerators, Journal of Analysis and Its Alications (ZAA), 2014, (33), 289 303. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical oints theory and alications, J. Funct. Anal., 1973, 04, 349 381. [3] C. O. Alves, J. M. do Ó, and O. H. Miyagaki, On a class of singular biharmonic roblems involving critical exonents, J. Math. Anal. Al., 2003, 277, 12 26. [4] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear ellitic equation of Kirchhoff tye, Comut. Math. Al., 2005, 49, 85 93. [5] C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal ellitic roblems with critical growth, Diff. Equa. Al., 2010, 2, 409 417. [6] G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order -Kirchhoff roblems, Commun. Contem. Math., 2014, 16(5), 43 ages.
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