Existence of Solutions for Semilinear Integro-differential Equations of p-kirchhoff Type

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1 Armenian Journal of Mathematics Volume 6, Number 2, 2014, Existence of Solutions for Semilinear Integro-differential Equations of p-kirchhoff Type E. Cabanillas Lapa *, W. Barahona M.**, B. Godoy T.*** and G. Rodriguez V. Instituto de Investigación, Facultad de Ciencias Matemáticas-UNMSM, Lima, Perú * cleugenio@yahoo.com ** wilbara 73@yahoo.es *** bgodoyt@unmsm.edu.pe grodriguezv@unmsm.edu.pe Abstract In our research we will study the existence of weak solutions to the problem [M u p 1,p)] p 1 p u = fx, u) + kx, y)hu)dy in, with zero Dirichlet boundary condition on a bounded smooth domain of R n, 1 < p < N; M,f,k and H are given functions. By means of the Galerkin method and using of the Brouwer Fixed Point theorem we get our results. The uniqueness of a weak solution is also considered. Key Words: p-kirchhoff type equations; variational methods; boundary value problems Mathematics Subject Classification 2000: 35J60, 35J35, 35J70, 35J65 Introduction The following equation ρ 2 u t 2 ρ0 h + E 2L L 0 u ) 2 2 u dx = 0, 1) x x2 presented by Kirchhoff in 1883 [13], is an extension of the classical D Alembert s wave equation by considering the effects of the changes in the length of the string during the vibrations. The parameters in 1) have the following meanings: L is the length of the string, h is the area 53

2 54 E. CABANILLAS L.,W. BARAHONA M., B. GODOY T. AND G. RODRIGUEZ V. of the cross-section, E is the Young modulus of the material, ρ is the mass density and P 0 is the initial tension. A distinguishing feature of equation 1) is that the equation contains a nonlocal coefficient ρ 0 + E L u h 2L 0 x 2 dx which depends on the average 1 L u 2L 0 x 2 dx, and hence the equation is no longer a pointwise identity. Some early classical studies of Kirchhoff equations were of Bernstein [7] and Pohožaev [18]. The equation ) a + b u 2 dx u = fx, u) in, u = 0 on, is related to the stationary analogue of the equation 1). Equation 1) received much attention only after Lions [20] proposed an abstract framework to the problem. Problems like 2) can be used for modelling several physical and biological systems where u describes a process which depends on the average of it self, such as the population density See [12] and its references therein). Some important and interesting results can be found, for example, in [3, 11, 16]. Recently Alves et al. [4] and Ma and Rivera [23] have obtained positive solutions of such problems by variational methods. An interesting generalization of problem 2) is [ p 1 p M u 1,p)] p u = fx, u) in, 2) u = 0, on, where is a bounded domain in R N with smooth boundary, and p u is the p-laplacian: p u := div u p 2 u), and. 1,p is the usual norm u p 1,p = u p dx in the Sobolev space W 1,p 0 ). Correa and Nascimento [8], Liu et al [24], Yang and Chang [25], Correa and Figueiredo [9], and more recently Molica Bisci and Radulescu [15] studied questions on the existence of positive solutions. In [25] Yang and Zhang studied the following problem [ p 1 p M u 1,p)] p u = λfx, u) in, u ν = 0, on, where p < N, they have established existence and multiplicity of solutions for the problem under suitable assumptions on M and f. In [10] Correa and Nascimento considered the following nonlocal elliptic system of p-kirchhoff type p 1 p [M 1 u 1,p)] p u = λfx, u) + h 1 x) in, p 1 p [M 2 v 1,p)] p v = λgx, v) + h 2 x) in, u ν = 0 v ν = 0, on, 3)

3 EXISTENCE OF SOLUTIONS FOR SEMILINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF P-KIRCHHOFF TYPE 55 with suitable hypotheses on M i, h i i = 1, 2), f and g.the authors have proved the existence of a weak solution for 3). In this paper we are interested in the following semilinear integro-differential equation of p-kirchhoff type [ p 1 p M u 1,p)] p u = fx, u) + with the following conditions: u = 0, on, kx, y)huy))dy in, M) the function M : R + R + is a continuous function and there is a constant m 0 > 0 such that Mt) m 0 for all t 0. F) fx, t) : R R is a continuous function and satisfies the subcritical condition 4) fx, t) c 1 t q 1 + 1) for some Np if N 3, p < q < p N p = + if N = 1, 2. H) H CR) satisfying Hs) c 2 s r, r 1; p 1. K) kx, y) is a non-positive L p ) function. The nonlocal term kx, y)hu)dy, with k = kx), appears in numerous physical models such as systems of particles in thermodynamical equilibrium via gravitational Coulomb) potential, 2-D fully turbulent behavior of real flow, thermal runaway in Ohmic Heating, shear bands in metal deformed under high strain rates, see [22] for references of these applications. Semilinear integro-differential equations have become an active area of research, for example in the framework of control theory as well in order to solve noncooperative system, arisen in the classical FitzHugh-Nagumo systems, see e.g. [2, 14, 5]. In case that the kernel k= kx,y) is symmetric and Hs) = s), the problem is of variational type and a solution can be found by the Mountain Pass Theorem if the L p L p norm is sufficiently small, see [6] for p = 2. Motivated by the above papers and the results in [14, 15], we consider 4) to study the existence of weak solutions, but with non-symmetric kernels, then the problem has no variational structure; so, the most usual variational techniques can not be used. To attack problem 4) we will use the Galerkin method through the following version of the Brouwer fixed -point theorem whose proof may be found in Lions see [21, lemma 4.3]).

4 56 E. CABANILLAS L.,W. BARAHONA M., B. GODOY T. AND G. RODRIGUEZ V. Proposition 1 Suppose that F : R m R m is a continuous function such that F ξ), ξ 0 on ξ = r, where, is the usual inner product in R m and is its related norm. Then, there exists z 0 B r 0) such that F z 0 ) = 0. 1 Main results and proofs Since we are preoccupied with the existence of weak solutions of the problem 4) we begin giving the definition of such solutions. Definition 1 A weak solution of problem 4) is any u W 1,p 0 ) such that [ )] p 1 M u p 1,p u p 2 u. v dx = fx, u)v dx + kx, y)huy)) dy)v dx for all v W 1,p 0 ). Our main result is given by the following theorem Theorem 1 Let us assume that conditions M) F) H) and K) hold. If k L p ) is sufficiently small and the function f satisfies fx, u)u a u p + b u 5) for some constants a, b > 0 with m p 1 0 aλ 1 1 k L p )c r rp c p c 2 > 0, = 1 p p where λ 1 is the biggest constant satisfying u p dx 1 u p dx; u 0, u W 1,p 0 ), λ 1 see [19]), and c ξ is the Sobolev constant for the embedding W 1,p 0 ) L ξ ) where ξ [ ; then problem 4) has at least one weak solution. Besides, any solution of 4) [1, Np N p satisfies the estimate u 1,p R 1 = max 1, b 1/p λ 1/p 1 m p 1 0 aλ 1 1 k L p )c r rp c p c 2 ) 1/p 1) Proof. Let {w ν } ν 1 be a Schauder s basis for W 1,p 0 ).For each m N consider the finite dimensional space V m = span{w 1,..., w m }. Since V m, ) and R m, ) are isometric and isomorphic, where is the usual norm in W 1,p 0 ) and is the Euclidian norm in R m, we make the identification u m = m ξ j w j ξ = ξ 1,..., ξ m ), j=1 u = ξ. 6)

5 EXISTENCE OF SOLUTIONS FOR SEMILINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF P-KIRCHHOFF TYPE 57 We will show that for each m there is u m V m, an approximate solution of 4), satisfying [ )] p 1 M u p 1,p u m p 2 u m. w j dx = fx, u m )w j dx+ kx, y)hu m )dy)w j dx, 7) j = 1, 2, 3,..., m. To solve this algebraic system in m unknowns ξ 1, ξ 2,..., ξ m, we consider the function F : R m R m given by F ξ) = F 1 ξ),..., F m ξ)), [ )] p 1 F j ξ) = M u p 1,p u p 2 u. w j dx fx, u)w j dx kx, y)huy))dy)w j dx where j = 1, 2, 3,..., m. and u V m We note that F is continuous from the continuity of M, fx, u) and kx, y)hu)dy, with respect to u. Therefore, from the hypotheses we have F u), u m p 1 0 u p 1,p a λ 1 u p 1,p b 1/q λ 1/p u 1,p k L p )c r rp c p c 2 u r+1 1,p m p 1 0 a λ 1 ) u p 1,p b 1/q λ 1/p u 1,p k L p )c r rp c p c 2 u r+1 1,p > 0 8) if u 1,p = R, for R large enough.thus, because of Proposition 1 there is u m V m, u m 1,p R, where R does not depend on m, such that u m is a solution of 7). Let us prove that the sequence u m ) m 1 W 1,p 0 ) has a convergent subsequence which converges to a solution of 4). Indeed, since u m ) is bounded, there exists a subsequence, still denoted by u m ), such that In view of continuity of M and the continuity of the Nemytskii map u p 1,p γ, for some γ, u m u, in W 1,p 0 ), 9) u m u, in L q ), 1 q < p, u m u, a.e in. [M u m p 1,p)] p 1 [Mγ)] p 1 10) f., u m ) f., u) in L q ). 11) But under the assumption K) we have kx, y) p = kx, y) p dy) 1 p < + ; i.e, kx, y) L p )

6 58 E. CABANILLAS L.,W. BARAHONA M., B. GODOY T. AND G. RODRIGUEZ V. for fixed x and noting that Hu m )) m 1 is bounded in L p ), 1 p + 1 p = 1, we obtain, up to a subsequence, that Hu m ) Hu) in L p ) 12) Therefore, for x we get kx, y)hu m y)) dy kx, y)huy)) dy, Also, we can easily prove that kx, y)hu m y)) dy L p ) < +. a.e. Then, by Lemma 1.3 in [21], we have kx, y)hu m y)) dy weakly in L p ). kx, y)huy)) dy 13) From the theory of monotone operators and reasoning similarly as in [8], we get [M u m p 1,p)] p 1 u m p 2 u m. w dx [Mγ)] p 1 u p 2 u w dx 14) w W 1,p 0 ). Now fixing l m, V l V m, letting m + in 7), and using 10) 14), we conclude that [Mγ))] p 1 u p 2 u w l dx = fx, u)w l dx + kx, y)hu)dy)w l dx, l = 1, 2,...15) From the completeness of {w ν } ν 1, the identity 15) holds with w l replaced by any w W 1,p 0 ). In particular, when w = u we get [Mγ))] p 1 u p 1,p = fx, u)u dx + kx, y)hu)dy)u dx. 16) On the other hand, taking w j = u m in 7) and passing to the limit, we obtain [Mγ))] p 1 γ = fx, u)udx + kx, y)hu)dy)udx. 17) Comparing equations 16) and 17) we get Then we conclude γ = u p 1,p. [Mγ)] p 1 γ = [Mγ)] p 1 u p 1,p.

7 EXISTENCE OF SOLUTIONS FOR SEMILINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF P-KIRCHHOFF TYPE 59 Therefore, from 15) with w l = w W 1,p 0 ) ) M u p 1,p)] p 1 u p 2 u w dx = fx, u)w dx + for all w W 1,p 0 ), which shows that u is a weak solution of 4). Finally, if u is any solution of 4), then M u p 1,p)] p 1 Therefore, either u 1,p 1 or u p dx = fx, u)u dx + kx, y)huy))dy)w dx 18) kx, y)huy))dy)u dx. m p 1 0 u p 1,p a λ 1 u p 1,p + b 1/p λ 1/p u 1,p + k L p )c r rp c p c 2 u r+1 1,p. 19) Then m p 1 0 aλ 1 1 k L p )c r rp c p c ) 2 u p 1,p b 1/p λ 1/p u 1,p 20) and 6) follows. 2 Uniqueness of weak solutions In this section, we are going to consider problem 4) when the exponent p satisfies 2N 2 + N < p 2. 21) To establish the uniqueness of weak solutions, we need the following lemma see [17]). Lemma 1 If p ]1, 2], then it holds i) z p 2 z y p 2 y β z y p 1 ii) z p 2 z y p 2 y, z y p 1) z y 2 z p + y p ) p 2 p for all y, z R N with β independent of y and z. Theorem 2 Let the assumptions of theorem 1) hold with 5) replaced by fx, u) fx, v)) u v) 0 x ; u, v R. 22) Let us assume, in addition, that M is Lipschitz on [0, R1], p where R 1 is defined in 6), and H is a C 1 -function such that H s) c 3 s r 1, c 3 > 0. Then if the Lipschitz constant L M of M is small enough, problem 4) has exactly one solution.

8 60 E. CABANILLAS L.,W. BARAHONA M., B. GODOY T. AND G. RODRIGUEZ V. Proof. We will follow some ideas in [1], adapted to our case. The part of existence follows from theorem 1. Now, let u 1 and u 2 be two solutions to the problem. Introduce the function u = u 1 u 2. Taking it for the test-function in the integral identities for u 1 and u 2, we obtain the relation [ )] p 1 ) M u 1 p 1,p u 1 p 2 u 1 u 2 p 2 u 2 u 1 u 2 ) dx = fx, u 1 ) fx, u 2 ))u 1 u 2 ) dx + kx, y)hu 1 y)) Hu 2 y)))u 1 u 2 ) dy dx { [ )] p 1 [ )] } p 1 + M u 2 p 1,p M u 1 p 1,p u 2 p 2 u 2 u 1 u 2 ) dx Now, using the hypotheses on M, H, K and Lemma 1, after some calculations we have ) p 2 m p 1 0 p 1) u 2 u 1 p + u 2 p p dx c 3 kx, y) u 1 r 1 + u 2 r 1) u 2 dy dx + βl p 1 M p [ u 1 1,p + u 2 1,p ) u 2 1,p ] p 1 u 2 1,p, where β = that 23) max s [0,R p 1 ] Ms). It follows from Holder s inequality and the Sobolev immersions kx, y) u 1 r 1 u 2 dy dx k L p u 1 r 1 r 1)p u 2 2p cr 1 r 1)p c 2 2p k L p u 1 r 1 1,p u 2 1,p where we take Similar inequality is obtained for u 2p p N ) 2. p Let us take a constant q, 1 2N )., 1 Using the inverse Holder s inequality for 2 2+N every constant q 0, 1) in 23), we obtain m p 1 ) q 1 g q q q 1 dx f q dx ) 1 q f g dx ) 1 ) 2 p 0 p 1) u 2q q dx u 1 p + u 2 p p. q ) q 1 1 q q dx ) 2c 3 c r 1 r 1)p c 2 2p k L prr p 3 R 2p 2 1 C β L M m p 2 1 u 2 1,p Since p < 2q < 2, again using Holder s inequality and the Sobolev embedding W 1,2q W 1,p and noting that 2 p. q p 1 q 1, we obtain m p 1 0 p 1)C2q 2 1 p 2 θ 2 2 R p u 1,p }{{} e 0 ) 2c 3 c r 1 r 1)p c 2 2p k L prr p 3 R 2p 2 1 C β L M m p 2 1 }{{} e 1 ) 2 u 1,p 24) 25)

9 EXISTENCE OF SOLUTIONS FOR SEMILINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF P-KIRCHHOFF TYPE 61 with θ = 1 2 p)q p1 q). Hence it follows that e 0 e 1 ) u 2 1,p = 0. 26) Therefore, if k L p and L M are small enough, we conclude that u 1,p = 0, and so u = 0. Acknowledgement The authors would like to express their deep thanks to the anonymous referees for their careful reading of the manuscript and invaluable suggestions for the revision of the manuscript. This work is partially supported by the Proyecto de Investigación N , UNMSM-FCM. References [1] S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions, Nonlinear Anal ), [2] C. O Alves and D.G Figuereido, Nonvariational elliptics systems via Galerkin methods, Funtion Spaces, differential Operators and non linear analysis, in the Hans Tridbel Anniversary VolumeBirkhauser Verlag 1. Switzerland, 2003), pp [3] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc ), pp [4] C. O. Alves, F. J. S. A. Corrêa, T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl ), pp [5] K. Balachandran, R. Sakthivel, Controllability of functional semilinear integrodifferential systems in Banach space, J.Math. Anal. Appl, ) n.2, pp [6] M. Biroli, M. Matzeu, S. Mataloni, Stability results for Mountain Pass and Linking type solutions of semilinear problems involving Dirichlet forms, NoDEA, ), Issue 3, pp [7] S. Bernstein, Sur une classe d équations fonctionnelles aux dérivées partielles, Bull. Acad. Sci. URSS. Ser. Math. [Izv. Akad. Nauk SSSR] ), pp [8] F. J. S. A. Corrêa, R.G. Nascimento, On the Existence of Solutions of a Nonlocal Elliptic Equation with a p-kirchhoff-type Term, Int.J.Math. Sc. 2008), Article ID, , doi: /2008/ [9] F. J. S. A. Corrêa, G. M. Figueiredo, On a Elliptic equation of p-kirchhoff type via variational methods, Bull. Austral. Math. Soc ), pp

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11 EXISTENCE OF SOLUTIONS FOR SEMILINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF P-KIRCHHOFF TYPE 63 [24] C. Liu, J. Wang and Q. Gao, Existence of nontrivial solutions for p-kirchhoff type equations, Bound.Val.Prob. 2013), 2013:279. [25] Y.Yang and J. Zhang, Existence results for a class of nonlocal problems involving p- Laplacian. Bound Val. Prob.2011), 2011:32.