MULTIPLE POSITIVE SOLUTIONS FOR A KIRCHHOFF TYPE PROBLEM

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1 Electronic Journal of Differential Equations, Vol. 205 (205, No. 29, pp. 0. ISSN: URL: or ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS FOR A KIRCHHOFF TYPE PROBLEM WENLI ZHANG Astract. In this article, we study the existence and multiplicity of positive solutions of a Kirchhoff type equation on a smooth ounded domain Ω R 3, and we show that the numer of positive solutions of the equation depends on the topological properties of the domain. The technique is ased on Ljusternik- Schnirelmann category and Morse theory.. Introduction This article concerns the multiplicity of positive solutions to the elliptic prolem ( ε 2 a + ε u 2 u + u = u p u, x Ω Ω (. u = 0, x Ω where Ω is a smooth ounded domain of R 3, ε > 0, a, > 0 are constants, 3 < p < 5. In recent years, some mathematicians considered the prolem ( a + u 2 u = f(x, u, x Ω Ω (.2 u = 0, x Ω where Ω R 3 is a smooth ounded domain. See for example[2, 9, 0, 4, 7, 8, 9, 20]. When a =, = 0, R 3 is replaced y R N, and u p u is replaced y f(u, Equation (. reduces to ε 2 u + u = f(u, u = 0, x Ω x Ω (.3 where Ω is a ounded domain of R N. Benci and Cerami [4] used Morse theory to estimate the numer of positive solutions of the prolem (.3. They proved that for ε sufficiently small the numer of positive solutions depends on the topology of Ω, actually on the Poincaré polynomial of Ω, P t (Ω, defined elow. Candela and Lazzo [8] considered the same equation with mixed Dirichlet-Neumann oundary conditions and f(t = t p 2 t. It was proved that the numer of positive solutions is 200 Mathematics Suject Classification. 35J50. Key words and phrases. Positive solutions; Kirchhoff type equation; Morse theory; Ljusternik-Schnirelmann category. c 205 Texas State University - San Marcos. Sumitted Septemer 2, 204. Pulished Novemer 23, 205.

2 2 WENLI ZHANG EJDE-205/29 influenced y the topology of the part Γ of the oundary Ω where ε is sufficiently small. Recently, Benci, Bonanno, Ghimenti and Micheletti [3, 3] proved that the numer of solutions of (.2 with f(t = t p 2 t on a smooth ounded domain Ω R 3 depends on the topological properties of the domain. More recently, Ghimenti and Micheletti [2] extended the result in [3, 3] to the Klein-Gordon-Maxwell and Schröedinger-Maxwell system and showed that the geometry of the 3-dimensional Riemannian manifold has effects on the numer of solutions of oth systems. Moreover, as far as we known, the existence and multiplicity of nontrivial solutions to the Kirchhoff equation have not ever een studied y using Morse theory. Motivated y the works descried aove and the fact, we will try to get the multiplicity of positive solutions to (. y using Ljusternik-Schnirelmann category and Morse theory. So in this paper we shall fill this gap. Our main results read as follows. Theorem.. Let 3 < p < 5. For ε > 0 sufficiently small, the prolem (. has at least cat(ω positive solutions. Theorem.2. Let 3 < p < 5. Assume that for ε > 0 sufficiently small all the solutions of the prolem (. are nondegenerate. Then there are at least 2P (Ω positive solutions, 2. Notation and preliminary results Throughout this article, we use the following norms for u H0 (Ω: ( /2, ( /p, u ε = ε 3 ε 2 u dx 2 u ε,p = Ω ε 3 u dx p ( /2, ( /p u = u 2 dx u p = u p dx R 3 R 3 and we denote y H ε the Hilert space H0 (Ω endowed with ε norm. Following the work y He and Zou [5], we let U(x e the positive ground state solution of ( a + u 2 u + u = u p u, x R 3 R 3 (2. u H (R 3, u(x > 0, x R 3 and I (U = m = inf u M I (u, where I (u = a 2 u u u 4 p + u+ p+ p+. M = {u H (R 3 \{0} : G (u = I (u, u = 0} For ε > 0 we set U ε (x = U(x/ε. Oviously U ε (x is the solution of the prolem ( ε 2 a + ε u 2 u + u = u p u, x Ω Ω u H 0 (Ω, u > 0, x Ω Now we shall recall some topological tools. Definition 2. ([6]. Let X a topological space and consider a closed suset A X. We say that A has category k relative to X(cat X (A = k if A is covered y k closed sets A i, i =, 2,..., k, which are contractile in X, and k is the minimum integer with this property. We simply denote cat(x = cat X (X.

3 EJDE-205/29 MULTIPLE POSITIVE SOLUTIONS 3 Remark 2.2 ([5]. Let X and X 2 e topological space. If g : X X 2 and g 2 : X 2 X are continuous operators such that g 2 g is homotopic to the identity on X, then cat(x cat(x 2. Definition 2.3. Let X is a topological space and let H k (X denotes its k-th homology group with coefficients in Q. The Poincaré polynomial P t (X of X is defined as the following power series in t, P t (X = k 0(dim H k (Xt k. If X is a compact space, we have that dim H k (X < and this series is finite. In the case P t (X is a polynomial and not a formal series. Remark 2.4 ([4]. Let X and Y e topological spaces. If f : X Y and g : Y X are continuous operators such that g f is homotopic to the identity on X, then P t (Y = P t (X + Z(t where Z(t is a polynomial with nonnegative coefficients. 3. Proof of main results To prove our main results, we consider the functional I ε C 2 (H ε, R, defined y I ε (u = a 2 u 2 ε + 2 u 2 ε,2 + 4 u 4 ε p + u+ p+ ε,p+ Oviously, there exists a one to one correspondence etween the nontrivial solutions of prolem (. and the nonzero critical points of I ε on H ε. As the functional I ε is not ounded elow on H ε, we introduce the manifold M ε = {u H ε \{0} : G ε (u = I ε(u, u = 0} Next, we present some properties of I ε and M ε. Lemma 3.. ( For any u H ε \{0}, there is a unique t ε > 0 such that u tε (x = t ε u(x M ε. (2 For any ε > 0, M ε is a C sumanifold of H ε, and there exists σ ε > 0 and K ε > 0 such that for any u M ε u ε σ ε, I ε (u K ε. (3 It holds (P S condition for the functional I ε on M ε. Proof. ( For any u H ε \{0} and t > 0, set u t (x = tu(x. Consider Υ ε (t = I ε (u t = a 2 t2 u 2 ε + 2 t2 u 2 ε,2 + 4 t4 u 4 ε p + tp+ u + p+ ε,p+. By computing, we known that Υ ε has a unique critical point t ε > 0 corresponding to its maximum. Then Υ ε (t ε = max t>0 Υ ε (t and Υ ε(t ε = 0. So G ε (u tε = 0 and u tε M ε. (2 By lemma 3. (, M ε. If u M ε, using that G ε (u = 0 and 3 < p < 5, we have G ε(u, u = a(2 (p + u 2 ε + (2 (p + u 2 ε,2 + (4 (p + u 4 ε < 0. So M ε is a C manifold. Using that G ε (u = 0 and the Soolev emedding, we have a u 2 ε + u 2 ε,2 + u 4 ε = u + p+ ε,p+ C u p+ ε,

4 4 WENLI ZHANG EJDE-205/29 a C u p ε. So the conclusion u ε σ ε follows. For any u M ε, I ε (u = a( 2 p + u 2 ε + ( 2 p + u 2 ε,2 + ( 4 p + u 4 ε. Using 3 < p < 5 and u ε σ ε, the conclusion I ε (u K ε follows. (3 Let {u n } is (P S sequence for I ε on M ε, that is I ε (u n c, I ε Mε (u n 0. Then it is easy to prove that u n ε is ounded. Going if necessary to a susequence, we can assume that u n 2 ε A(> 0, u n u in H ε, u n u in L s (Ω ( s < 6. Oviously, we have ρ ε (u = a u 2 ε + u 2 ε,2 + A u 2 ε u + p+ ε,p+ = 0. Set ω n = u n u. By Brézis-Lie Lemma, we have ω n 2 ε = u n 2 ε u 2 ε + o n (. Since I ε(u n, u n = o n (, we otain (a + A ω n 2 ε + ρ ε (u = o n (. This concludes the proof. By Lemma 3. (2, we otain I ε Mε is ounded from elow. By using Lagrange multiplier method, we known that M ε contains every nonzero solution of prolem (., and define the minimax m ε as m ε = inf u M ε I ε (u Proof of Theorem.. Since the functional I ε C 2 is ounded elow and satisfies the (PS condition on the complete manifold M ε, we have, y the classical Ljusternik-Schnirelmann category result [7], that I ε has at least cati d ε critical points in the sulevel I d ε = {u H ε : I ε (u d} In the following, we will prove that, for ε and δ sufficiently small, it holds cat(ω cat(m ε I m +δ ε To prove this, we uild two continuous functions where Φ ε : Ω M ε I m +δ ε, (3. β : M ε I m +δ ε Ω +, (3.2 Ω = {x Ω : d(x, Ω < r}, Ω + = {x R 3 : d(x, Ω < r} with r > 0 small enough so that cat(ω = cat(ω + = cat(ω. Following the idea in [6], we can find two functions Φ ε and β such that β Φ ε : Ω Ω + is homotopic to the immersion i : Ω Ω +. By Remark 2.2 we otain the inequality which completes the proof.

5 EJDE-205/29 MULTIPLE POSITIVE SOLUTIONS 5 Proof of Theorem.2. By Remark 2.4, (3. and (3.2, we have P t (M ε Iε m +δ = P t (Ω + Z(t where Z(t is a polynomial with nonnegative coefficients. Since inf ε m ε = c > 0, we have By Morse theory we have P t (I m +δ ε, I c 2 ε = tp t (M ε I m +δ ε, (3.3 P t (H ε, Iε m +δ = t(p t (Iε m +δ, I c 2 ε t. (3.4 t µ(u = P t (H ε, Iε m +δ u K ε + P t (I m +δ ε, I c 2 ε + ( + tq ε (t (3.5 where K ε e the set of critical points of I ε, µ(u is the Morse index of u, Q ε (t is a polynomial with nonnegative coefficients. Using this relation with (3.3 (3.5, we otain t µ(u = tp t (Ω + t 2 (P t (Ω + t( + tq ε (t (3.6 u K ε Theorem.2 easily follows y evaluating the power series (3.6 for t =. For ξ Ω we define the function 4. The function Φ ε ω ξ,ε (x = U ε (x ξχ r ( x ξ (4. where χ r is a smooth cut off function χ r for t [0, r 2, χ r 0 for t > r and χ r(t 2/r. We define Φ ε : Ω M ε y Φ ε (ξ = t ε (ω ξ,ε ω ξ,ε (x Remark 4.. We have that the following limits hold uniformly with respect to ξ Ω, ω ξ,ε ε U, ω ξ,ε ε,p U p. Proposition 4.2. For any ε > 0 the map Φ ε is continuous. Moreover for any δ > 0 there exists ε 0 > 0 such that if ε < ε 0 then I ε (Φ ε (ξ < m + δ. Proof. Oviously, Φ ε is continuous. We claim that t ε (ω ξ,ε uniformly with respect to ξ Ω. In fact, y Lemma 3. t ε (ω ξ,ε is the unique solution of at ω ξ,ε 2 ε + t ω ξ,ε 2 ε,2 + t 3 ω ξ,ε 4 ε = t p ω + ξ,ε p+ ε,p+ By Remark 4. we have the claim. Now, y Remark 4. and the aove claim we have I ε (Φ ε (ξ = ( 2 p + at2 ε ω ξ,ε 2 ε + ( 2 p + t2 ε ω ξ,ε 2 ε,2 + ( 4 p + t4 ε ω ξ,ε 4 ε ( 2 p + a U 2 + ( 2 p + U 2 + ( 4 p + U 4 = m this completes the proof. Remark 4.3. Note that lim sup ε 0 m ε m.

6 6 WENLI ZHANG EJDE-205/29 5. The map β For u M ε we can define a point β(u R 3 y Ω β(u = x u+ p+ dx Ω u+ p+ dx The function β is well defined in M ε since if u M ε then u + 0. We shall prove that if u M ε Iε m +δ then β(u Ω +. First of all we consider partitions of the compact manifold Ω. Given ε > 0, a finite partition P ε = {Pj ε} j Λ ε is called a good partition if: for any j Λ ε the set Pj ε is closed; Pi ε Pj ε Pi ε P j ε for i j; there exist r (ε, r 2 (ε > 0 such that, for any j, there exists a point qj ε P j ε such that B(q ε j, ε P ε j B(q ε j, r 2 (ε B(q ε j, r (ε, with r (ε r 2 (ε Cε for some positive constant C; lastly, there exists a finite numer ι N such that every x Ω is contained in at most ι alls B(q ε j, r (ε, where ι does not depends on ε. Lemma 5.. There exists γ > 0 such that, for any δ > 0 and any ε (0, ε 0 (δ where ε 0 (δ is as in Proposition 4.2, given any good partition P ε of the domain Ω and for any u M ε Iε m +δ there exists a set Pj ε such that ε 3 u + p+ dx γ Proof. Taking into account that G ε (u = 0 we have a u 2 ε u + p+ ε,p+ = ε 3 u + p+ dx j Pj ε P ε j j u + j p+ ε,p+ = j u + j p ε,p+ u+ j 2 ε,p+ max{ u + j j p ε,p+ } j u + j 2 ε,p+ where u + j is the restriction of the function u + on the set Pj ε. Arguing as in [3], we prove that there exists a constant C > 0 such that u + j 2 ε,p+ Cι u + 2 ε, thus that concludes the proof. j max{ u + j j p ε,p+ } a Cι Proposition 5.2. For any η (0, there exists δ 0 > 0 such that for any δ (0, δ 0 and any ε (0, ε 0 (δ as in Proposition 4.2, for any u M ε Iε m +δ there exists a point q = q(u Ω such that ε 3 u + p+ 2(p + ( dx > ( η 4m + 2a2 2a a 2 + 3m B(q, r 2 p

7 EJDE-205/29 MULTIPLE POSITIVE SOLUTIONS 7 Proof. We prove only the proposition for any u M ε Iε mε+2δ. Indeed, y this result and y Remark 4.3 we get lim ε 0 m ε = m Hence it holds Iε m +δ Iε mε+2δ for δ, ε small enough. So the thesis holds. We argue y contradiction. Suppose that there exists η (0, such that we can find vanishing sequences {δ k }, {ε k } and a sequence {u k } M εk I mε k +2δ k ε that m εk I εk (u k = ( 2 p + a u k 2 ε k + ( 2 p + u k 2 ε k,2 + ( 4 p + u k 4 ε k m εk + 2δ k m + 3δ k. such (5. for k large enough, and for any q Ω, u + 2(p + B(q, r 2 k p+ dx ( η p (4m + 2a2 2a a2 + 3m. (5.2 ε 3 k By Ekeland variational principle and y definition of M εk we can assume that I ε k (u k 0. (5.3 By Lemma 5., there exists a set P ε k k P εk such that ε 3 u + k p+ dx γ. (5.4 So we can choose a point q k P ε k k P ε k k, and define, for z Ω ε k = ε k (Ω q k, ω k (z = u k (ε k z + q k = u k (x, where x Ω. We otain that ω k H 0 (Ω εk. By (5., we have ω k 2 H 0 (Ωε k C. So we otain ω k ω in Hloc (R3, ω k ω in L s loc (R3 (2 s < 6 and ω k 2 A. Thus we prove that ω 0 and A > 0 y (5.4. Next we claim dist(q k, Ω lim =. (5.5 k ε k We argue y contradiction. Suppose that It is easy to verify that ω is a solution of dist(q k, Ω lim = d <. k ε k (a + A u + u = u p u, x R 3 + u(x = 0, x R 3 + (5.6 where R 3 + is a half space. We know that (5.6 has no nontrivial solution from the work y [, ]. So ω 0, this contradicts with ω 0. This concludes the claim. By (5.5, Ω εk converges to the whole space R 3 as k. Using (5. (5.3 and computing, we have I (ω k m, I (ω k 0. (5.7

8 8 WENLI ZHANG EJDE-205/29 This implies {ω k } M is a minimizing sequences for m. Arguing as in [5], we have ω k ω, ω M, I (ω = m and I (ω = 0. By using (5.7 and Pohozaev identity, we have Thus, we otain that B(0,T a 2 ω k ω k ω k 4 p + ω k p+ p+ = m + o k (, a ω k 2 + ω k ω k 4 ω k p+ p+ = o k( a 2 ω k ω k ω k 4 3 p + ω k p+ p+ = o k( ω k p+ p+ 2(p + ( p 4m + 2a2 2a a 2 + 3m So y ω k ω, for T and k large enough, we have ω + 2(p + ( k p+ dz > ( η 4m + 2a2 2a a 2 + 3m p On the other hand y (5.2 and the definition of ω k, for any T > 0 we have, for k large enough, ω + k p+ dz B(0,T ε 3 u k p+ dx k B(q k,ε k T ε 3 u k p+ dx k B(q k, r 2 2(p + ( η p (4m + 2a2 2a a2 + 3m. This leads to a contradiction. Proposition 5.3. There exists δ 0 > 0 such that for any δ (0, δ 0 and any ε (0, ε(δ 0 as in Proposition 5.2, for any u M ε Iε m +δ it holds β(u Ω +. Moreover the composition β Φ ε : Ω Ω + is homotopic to the immersion i : Ω Ω +. Proof. Arguing y contradiction, we suppose that there exist sequences {δ k }, {ε k } R and {u k } M εk I m +δ k ε such that δ k, ε k 0 +, as k, and β(u k Ω + for all k. By Ekeland variational principle and y definition of M εk we can assume that I ε k (u k 0. So y Proposition 5.2 we can find q k Ω such that ε 3 B(q k k, r 2 u k p+ 2(p+ ( dx ( η p 4m + 2a2 2a a 2 +3m ε 3 Ω u > k p+ dx 2(p+ ( k p 4(m + δ k + 2a2 2a a 2 +3(m +δ k Finally, β(u k q k ε 3 Ω (x q k u k p+ dx k Ω u k p+ dx ε 3 k

9 EJDE-205/29 MULTIPLE POSITIVE SOLUTIONS 9 ε 3 B(q k k, r (x q k u k p+ dx 2 ε 3 k Ω u k p+ dx r diam(ω ( + ε 3 Ω\B(q k k, r (x q k u k p+ dx 2 ε 3 k Ω u k p+ dx 2(p+ ( η p (4m + 2a2 2a a 2 +3m 2(p+ p (4(m + δ k + 2a2 2a a 2 +3(m +δ k The aove expression implies that β(u k Ω +, which contradicts β(u k Ω +. Acknowledgments. This work was supported y the Shanxi University Technology research and development (project References [] J. Ai, X. P. Zhu; Positive solutions of inhomogeneous elliptic oundary value prolems in the half space, Comm. Partial Differential Equations 5 (990, [2] C. O. Alves, F. J. S. A. Corrêa, T. F. Ma; Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005, [3] V. Benci, C. Bonanno, A. M. Micheletti; On the multiplicity of a nonlinear elliptic prolem on Riemannian manifolds, J. Funct. Anal. 252 (2007, [4] V. Benci, G. Cerami; Multiple positive solutions of some elliptic prolems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations 2 (994, [5] V. Benci, G. Cerami; The effect of the domain topology on the numer of positive solutions of nonlinear elliptic prolems, Arch. Ration. Mech. Anal. 4 ( [6] V. Benci, G. Cerami; The effect of the domain topology on the numer of positive solutions of nonlinear elliptic prolems, Arch. Rationa. Mech. Anal. 4( [7] V. Benci, G. Cerami, D. Passaseo; On the numer of the positive solutions of some nonlinear elliptic prolems, in Nonlinear Analysis, A triute in Honour of G Prodi, Quaderno Scuola Norm Sup, Pisa, 99, pp [8] A. Candela, M. Lazzo; Positive solutions for a mixed oundary prolem, Nonlinear Anal. 24 (995, [9] C. Chen, Y. Kuo, T. Wu; The Nehari manifold for a Kirchhoff type prolem involving signchanging weight functions, J. Diff. Equs. 250 (20, [0] B. Cheng, X. Wu; Existence results of positive solutions of Kirchhoff type prolems, Nonlinear Anal. Theory Methods Appl. 7 (2009, [] M. J. Estean, P. L. Lions; Existence and nonexistence results for semilinear elliptic prolems in unounded domains, Proc. Roy. Soc. Edim. 93 (982, -4. [2] M. Ghimenti, A. M. Micheletti; The role of the scalar curvature in some singularly pertured coupled elliptic systems on Riemannian manifolds, Discrete Contin. Dynam. Systems 34 (204, [3] M. Ghimenti, A. M. Micheletti; Positive solutions for singularly pertured nonlinear elliptic prolem on mainifolds via Morse theory, arxiv: (200. [4] X. He, W. Zou; Multiplicity of solutions for a class of Kirchhoff type prolems, Acta Math. Appl. Sin. Engl. Ser. 26 (200, [5] X. He, W. Zou; Existence and concentration ehavior of positive solutions for a Kirchhoff equations in R 3, J. Diff. Equs. 252 (202, [6] L. Ljusternik, L. Schnirelmann; Méthodes topologiques dans les prolèmes variationelles, Actualités Sci. Indust. 88 (934. [7] A. Mao, Z. Zhang; Sign-changing and multiple solutions of Kirchhoff type prolems without the (PS condition, Nonlinear Anal. 70 (2009, [8] K. Perera, Z. Zhang; Nontrivial solutions of Kirchhoff-type prolems via the Yang index, J. Diff. Equs. 22 (2006, [9] Y. Yang, J. Zhang; Nontrivial solutions of a class of nonlocal prolems via local linking theory, Appl. Math. Lett. 23 (200, [20] Z. Zhang, K. Perera; Sign changing solutions of Kirchhoff type prolems via invariant sets of descent flow, J. Math. Anal. Appl. 37 (2006,

10 0 WENLI ZHANG EJDE-205/29 Wenli Zhang Department of Mathematics, Changzhi University, Shanxi 0460, China address:

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