INFINITELY MANY SOLUTIONS FOR KIRCHHOFF-TYPE PROBLEMS DEPENDING ON A PARAMETER

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1 Electronic Journal of Differential Equations, Vol. 016 (016, No. 4, pp ISSN: URL: or INFINITELY MANY SOLUTIONS FOR KIRCHHOFF-TYPE PROBLEMS DEPENDING ON A PARAMETER JUNTAO SUN, YONGBAO JI, TSUNG-FANG WU Abstract. In this article, we study a Kirchhoff type problem with a positive parameter λ, Z K u dx u = λf(x, u, in, u = 0, on, where K : [0, + R is a continuous function and f : R R is a L 1 -Carathéodory function. Under suitable assumptions on K(t and f(x, u, we obtain the existence of infinitely many solutions depending on the real parameter λ. Unlike most other papers, we do not require any symmetric condition on the nonlinear term f(x, u. Our proof is based on variational methods. 1. Introduction Consider the Kirchhoff type problem ( K u dx u = λf(x, u, in, u = 0, on, (1.1 where λ > 0 is a real parameter, K : [0, + R is a continuous function, R N (N 1 is a nonempty bounded open set with a smooth boundary, f : R R is a L 1 -Carathéodory function. If K(t = a + bt, then (1.1 is related to the stationary case of equations that arise in the study of string or membrane vibrations, namely, ( u tt a + b u dx u = g(x, u, (1. where u denotes the displacement, g(x, u the external force and b the initial tension while a is related to the intrinsic properties of the string, such as Young s modulus. Equations of this type were suggested by Kirchhoff [16] in 1883 to describe the transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations. Equation (1. is often referred to as being nonlocal because of the presence of the integral over the entire domain. 010 Mathematics Subject Classification. 35B09, 35J0. Key words and phrases. Infinitely many solutions; Kirchhoff type problem; variational method. c 016 Texas State University. Submitted November 11, 015. Published August 17,

2 J. SUN, Y. JI, T.-F. WU EJDE-016/4 We wish to point out that similar nonlocal problems also model several physical and biological systems. For example, a parabolic version of (1. can be used to describe the growth and movement of a particular species theoretically. The movement, simulated by the integral term, is assumed dependent on the energy of the entire system with u as its population density. Alternatively, the movement of a particular species may be subject to the total population density within the domain (for instance, the spreading of bacteria which induces equations of the type ( u t a u dx u = g(x, u. Chipot-Lovat [1] and Corrêa [14] studied the existence of solutions and their uniqueness for such nonlocal problems and their corresponding elliptic equations. Since Lions [19] introduced an abstract framework of (1., the solvability of (1. has been well-studied in general dimensions and domains by various researchers [3, 5, 9]. More precisely, Arosio-Panizzi [5] studied the Cauchy-Dirichlet type problem related to (1. in the Hadamard sense as a special case of an abstract second-order Cauchy problem in a Hilbert space. D Ancona-Spagnolo [3] obtained the existence of a global classical periodic solution for the degenerate Kirchhoff equation with real analytic data. Compared with (1., its stationary equation, including both the case of bounded domain and the case of unbounded domain, has received more attention. We refer to [1,, 4, 6, 7, 10, 11, 13, 15, 17, 18, 0, 1,, 5, 6, 7, 8, 9] and the references therein. The research of these papers mainly involve the existence of positive solutions, ground state solutions and multiplicity of positive solutions under various assumptions on K(t and f(x, u. Let us briefly comment some known results related to our paper. Alves-Corrêa-Ma [] studied (1.1 with λ = 1 and found the conditions of K(t and f(x, u that permit the existence of a positive solution. That is, K(t does not grow too fast in a suitable interval near zero and f(x, u is locally Lipschitz subject to some prescribed criteria. Chen-Kuo-Wu [10] studied the following Kirchhoff type problem with concave and convex nonlinearities ( K u dx u = µf(x u q u + g(x u p u in, (1.3 u = 0 in, where is a smooth bounded domain in R N with 1 < q < < p < ( = N N if N 3, = if N = 1,, K(t = a + bt, a, b, µ > 0 are parameters and the weight functions f, g C( are allowed to be changing-sign. Using the Nehari manifold method and fibering map, several results on the existence and multiplicity of positive solutions for (1.3 are obtained. It is worth noting that they illustrated the difference in the solution behavior which arises from the consideration of the nonlocal effect. Ricceri [5] investigated the Kirchhoff type problem with two parameters ( K u dx u = λf(x, u + µg(x, u, in, (1.4 u = 0, on,

3 EJDE-016/4 INFINITELY MANY SOLUTIONS 3 Under rather general assumptions on K(t and f(x, u, he proved that for each λ > λ > 0 and for each Carathéodory function g(x, u with a sub-critical growth, (1.4 admits at least three weak solutions for every µ 0 small enough. Later, using a recent three critical points theorem due to Ricceri [4], Graef-Heidarkhani- Kong [15] improved the results in [5] and also obtained the existence of three weak solutions for (1.4 under some appropriate hypotheses, depending on two real parameters. Motivated by the above works, in the present paper we shall establish some results on the existence of infinitely many solutions for (1.1. In our results neither symmetric nor monotonic condition on the nonlinear term is assumed. We require that f(x, u have a suitable oscillating behavior either at infinity or at zero on u. For the first case, we obtain an unbounded sequence of solutions; for the second case, we get a sequence of non-zero solutions strongly converging at zero. It is worth emphasizing that we extend and improve the results in [17]. The remainder of this paper is organized as follows. In Section, some preliminary results are introduced. The main results and their proofs are presented in Section 3.. Preliminaries Let H0 1 ( be the usual Sobolev space endowed with norm ( 1/. u = u dx Throughout this paper, we denote the best Sobolev constant by S r for the imbedding of H 1 0 ( into L r ( with r < and define it by S r = u inf, u H0 1(\{0} u r where r stands for the classical L r norm. We recall that f : R R is an L 1 -Carathéodory function if (a the mapping x f(x, u is measurable for every u R; (b the mapping u f(x, u is continuous for almost every x ; (c for every ρ > 0 there exists a function l ρ L 1 ( such that f(x, u l ρ (x u ρ for almost every x. We shall prove our results by applying the following smooth version of [8, Theorem.1], which is a more precise version of Ricceri s Variational Principle [3, Lemma.5]. Theorem.1. Let E be a reflexive real Banach space, let Φ, Ψ : E R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous and coercive, and Ψ is sequentially weakly upper semi-continuous. For every r > inf E Φ, let ϕ(r := inf ( v Φ 1 (,r Ψ(v Ψ(u, r Φ(u γ := lim inf ϕ(r, δ := lim inf ϕ(r. r + r (inf E Φ +

4 4 J. SUN, Y. JI, T.-F. WU EJDE-016/4 Then the following properties hold: (a For every r > inf E Φ and every λ (0, 1/ϕ(r; the restriction of the functional I λ := Φ λψ to Φ 1 (, r admits a global minimum, which is a critical point (local minimum of I λ in E. (b If γ < + ; then for each λ (0, 1/γ, the following alternative holds: either (b1 I λ possesses a global minimum, or (b there is a sequence {u n } of critical points (local minima of I λ such that lim Φ(u n = +. (c If δ < + ; then for each λ (0, 1/δ, the following alternative holds: either (c1 there is a global minimum of Φ which is a local minimum of I λ, or (c there is a sequence {u n } of pairwise distinct critical points (local minima of I λ that converges weakly to a global minimum of Φ. 3. Main results The following theorem is our first result. Theorem 3.1. Let K : [0, + R be a continuous function and let f : R R be an L 1 -Carathéodory function. Assume that the following conditions hold: (A1 there exists a constant m > 0 such that inf t 0 K(t m; (A there exist constants m 1 > 0 and m 0 such that K(t m 1 t + m, for every t [0, +, where K(t = t K(sds for t 0; 0 (A3 there exist two constants a 1 > 0, a 0 and 1 < α < such that f(x, u a 1 u + a u α 1, for all x and u R; (A4 there exist x 0 and three constants τ > σ > 0 and such that where γ > a 1m 1 ( B(x 0, τ B(x 0, σ ms (τ σ B(x 0, σ B(x 0, σ B(x 0, τ, F (x, u 0, for all (x, u ( \ B(x 0, σ R, F (x, u γu, F (x, u = u 0 for all (x, u B(x 0, σ [1, +, f(x, sds, for all (x, u R, > 0, (3.1 and B(x 0, σ denotes the open ball with center at x 0 and radius σ. Then for every λ ( m 1 ( B(x 0, τ B(x 0, σ γ(τ σ, ms, B(x 0, σ a 1 Equation (1.1 has a sequence of solutions {u n } in H0 1 ( satisfying lim u n = +.

5 EJDE-016/4 INFINITELY MANY SOLUTIONS 5 Proof. Let Φ, Ψ : H0 1 ( R be defined by Φ(u = 1 K( u, Ψ(u = and put I λ (u := Φ(u λψ(u F (x, udx (3. for all u H 1 0 (. Using the properties of f, it is easy to verify that Φ, Ψ C 1 (H0 1 (, R and for any v H0 1 (, we have ( Φ (u, v = K u(x dx u(x v(xdx, Ψ (u, v = f(x, u(xv(xdx. So by the standard arguments, we deduce that the critical points of the functional I λ are the weak solutions of (1.1. Using (A1 and (3. gives Φ(u m u for all u H 1 0 (, (3.3 which implies that Φ is coercive. Moreover, it is easy to show that Φ is sequentially weakly lower semi-continuous and Ψ is sequentially weakly upper semi-continuous. Therefore, the functionals Φ and Ψ satisfy the regularity assumptions of Theorem.1. Obviously, it follows from (3.1 that Now we fix m 1 ( B(x 0, τ B(x 0, σ γ(τ σ B(x 0, σ Then for any r > 0, using (3.3 leads to and combining (A3, we have Thus, Ψ(u < ms a 1. λ ( m 1 ( B(x 0, τ B(x 0, σ γ(τ σ, ms. B(x 0, σ a 1 Φ 1 (, r = {u H 1 0 ( : Φ(u < r} ϕ(r = { u H 1 0 ( : u < r/m }, ( a1 ( a1 S F (x, u(x dx < a 1r ms + a αsα ( r m α/. u(x dx + a α u + a αsα u α inf ( v Φ 1 (,r Ψ(v Ψ(u r Φ(u u(x α dx (3.4

6 6 J. SUN, Y. JI, T.-F. WU EJDE-016/4 which implies Ψ(u r < a 1 ms + a αsα ( m α/ r α, γ := lim inf ϕ(r a 1 r + ms < +. Now we choose a sequence {η n } of positive numbers satisfying lim η n = +. For every n N, we define v n given by 0, x \ B(x 0, τ, η v n (x := n τ σ (τ dist(x, x 0, x B(x 0, τ \ B(x 0, σ, (3.5 η n, x B(x 0, σ. It is easy to verify that v n H0 1 ( and v n = v n dx + = = \B(x 0,τ B(x 0,τ\ η n (τ σ dx B(x 0,τ\ η n (τ σ ( B(x 0, τ B(x 0, σ. Then it follows from (A and (3. that Φ(v n m 1 v n + m v n dx + v n dx m 1η n( B(x 0, τ B(x 0, σ (τ σ + m. (3.6 On the other hand, using (A4, from the definition of Ψ, we infer that Ψ(v n F (x, η n dx. (3.7 Thus, by (3.6, (3.7 and (A4, for every n N large enough, one has I λ (v n m 1ηn( B(x 0, τ B(x 0, σ (τ σ + m λ F (x, η n dx < m 1ηn( B(x 0, τ B(x 0, σ (τ σ + m λ γηndx Since = m 1ηn( B(x 0, τ B(x 0, σ (τ σ + m λγ B(x 0, σ ηn = m + ( m1 ( B(x 0, τ B(x 0, σ (τ σ λγ B(x 0, σ λ > m 1( B(x 0, τ B(x 0, σ γ(τ σ B(x 0, σ and lim η n = +, we have lim I λ(v n =. ηn.

7 EJDE-016/4 INFINITELY MANY SOLUTIONS 7 This shows that the functional I λ is unbounded from below, and it follows that I λ has no global minimum. Therefore, by Theorem.1 (b, there exists a sequence {u n } of critical points of I λ such that lim u n = +, since K( u n m1 u n + m by (A. Therefore, the conclusion is achieved. Remark 3.. Indeed, from condition (A, we can see that the potential K has a sublinear growth. Moreover, it is not difficult to find such continuous function K satisfying (A1 and (A, for example Now, we state our second result. K(t = t, t 0. Theorem 3.3. Let K : [0, + R be a continuous function and f : R R be an L 1 -Carathéodory function. Assume that (A1 and the following conditions hold: (A there exist constants l 1 0, q 1 and l > 0 such that K(t l 1 t q 1 + l, for every t [0, + ; (A3 there exist b 1 > 0, b 0 and < β < ( = N N if N 3, = if N = 1, such that f(x, u b 1 u + b u β 1, for all x and u R; (A4 there exist x 0 and three constants τ > σ > 0 and such that γ > b 1l ( B(x 0, τ B(x 0, σ ms (τ σ B(x 0, σ B(x 0, σ B(x 0, τ, F (x, u 0, for all (x, u (\, B(x 0, σ R, F (x, u γu, for all (x, u B(x 0, σ [0, 1], > 0, (3.8 where B(x 0, σ denotes the open ball with center at x 0 and radius σ. Then for every λ ( l ( B(x 0, τ B(x 0, σ γ(τ σ, ms, B(x 0, σ b 1 Equation (1.1 has a sequence of solutions, which converges strongly to zero in H 1 0 (. Proof. It follows from (3.8 that Now we fix λ l ( B(x 0, τ B(x 0, σ γ(τ σ B(x 0, σ < ms b 1. ( l ( B(x 0, τ B(x 0, σ γ(τ σ, ms. B(x 0, σ b 1

8 8 J. SUN, Y. JI, T.-F. WU EJDE-016/4 Using the condition (A3 and (3.4, one has Ψ(u F (x, u(x dx Thus, there holds which implies [ b1 u(x dx + b β ( b 1 S u + b βs β u β β < b 1r ms + b βs β ( r m β/. β ( v Φ ϕ(r = inf 1 (,r Ψ(v Ψ(u r Φ(u Ψ(u r < b 1 ms + b βs β ( m β/ r β, β ] u(x β dx (3.9 δ := lim inf ϕ(r b 1 r 0 + ms < +. Now we choose a sequence {η n } of positive numbers satisfying lim η n = 0. For all n N, let v n defined by (3.5 with the above η n. Then, using (A and (A4, for every n N large enough one has I λ (v n l 1ηn q ( B(x 0, τ B(x 0, σ q q(τ σ q λ F (x, η n dx < l 1ηn q ( B(x 0, τ B(x 0, σ q q(τ σ q λ γηndx + l η n( B(x 0, τ B(x 0, σ (τ σ + l η n( B(x 0, τ B(x 0, σ (τ σ = l 1ηn q ( B(x 0, τ B(x 0, σ q q(τ σ q + l ηn( B(x 0, τ B(x 0, σ (τ σ λγ B(x 0, σ η n = l 1η q n ( B(x 0, τ B(x 0, σ q q(τ σ q + ( l ( B(x 0, τ B(x 0, σ (τ σ λγ B(x 0, σ η n < 0. Thus, lim I λ(v n = I λ (0 = 0, which shows that zero is not a local minimum of I λ. This, together with the fact that zero is the only global minimum of Φ, we deduce that the energy functional

9 EJDE-016/4 INFINITELY MANY SOLUTIONS 9 I λ does not have a local minimum at the unique global minimum of Φ. Therefore, by Theorem.1 (c, there exists a sequence {u n } of critical points of I λ, which converges weakly to zero. The proof is complete. Remark 3.4. Now we give an example to illustrate Theorem 3.3. Consider the Kirchhoff equation ( a + b u(x dx u = λf(x, u, in, (3.10 u = 0, on, where a > 0, b 0 and R N is bounded domain. Set K(t = a + bt. Obviously, (A1 and (A are satisfied. Let and α L ( with f(x, u = α(xu, inf α(x > x α(x( B(x 0, τ B(x 0, σ x S (τ, σ B(x 0, σ where x 0 is a point of and two constants τ > σ > 0 satisfying B(x 0, σ B(x 0, τ. Thus, (A3 and (A4 hold. Therefore, by Theorem 3.3, Equation (3.10 has infinitely many nontrivial solutions for every ( a( B(x 0, τ B(x 0, σ λ inf x α(x(τ σ B(x 0, σ, as. x α(x Acknowledgements. J. Sun is ported by the National Natural Science Foundation of China (Grant Nos and , Shandong Provincial Natural Science Foundation Grant (No. ZR015JL00, China Postdoctoral Science Foundation (Grant Nos. 014M and 015T80491, and Young Faculty Support Program of Shandong University of Technology. T.F. Wu was ported in part by the Ministry of Science and Technology, Taiwan (Grant M MY and the National Center for Theoretical Sciences, Taiwan. References [1] G. A. Afrouzi, N. T. Chung, S. Shakeri; Existence of positive solutions for Kirchhoff type equations, Electron. J. Differential Equations, Vol. 013 (013, No. 180, pp [] C. O. Alves, F. S. J. A. Corrêa, T. F. Ma; Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (005, [3] P. D Ancona, S. Spagnolo; Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (199, [4] S. Aouaoui; Existence of three solutions for some equation of Kirchhoff type involving variable exponents, Appl. Math. Comput. 18 (01, [5] A. Arosio, S. Panizzi; On the well-posedness of the Kirchhoff string. Trans. Amer. Math. Soc. 348 (1996, [6] G. Autuori, F. Colasuonno, P. Pucci; On the existence of stationary solutions for higher order p-kirchhoff problems, Commun. Contemp. Math. 16 (014 (43 pages. [7] A. Bensedki, M. Bouchekif; On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity, Math. Comput. Modelling 49 (009, [8] G. Bonanno, B. Di Bella; Infinitely many solutions for a fourth-order elastic beam equation, Nonlinear Differ. Equ. Appl. 18 (011,

10 10 J. SUN, Y. JI, T.-F. WU EJDE-016/4 [9] M. M. Cavalcanti, V. N. D. Cavalcanti, J. A. Soriano; Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (001, [10] C. Chen, Y. Kuo, T. F. Wu; The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 50 ( [11] B. Cheng, X. Xu; Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71 ( [1] M. Chipot, B. Lovat; Some remarks on non local elliptic and parabolic problems, Nonlinear Anal. 30 (1997, [13] F. Colasuonno, P. Pucci; Multiplicity of solutions for p(x-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 ( [14] F. J. S. A. Corrêa; On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal. 59 (004, [15] J. R. Graef, S. Heidarkhani, L. Kong; Variational approach to a Kirchhoff-type problem involving two parameters, Results. Math. 63 (013, [16] G. Kirchhoff; Vorlesungen uber mathematische Physik: Mechanik. Teubner, Leipzig (1883. [17] X. He, W. Zou; Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Analysis, 70 (009, [18] Y. Li, F. Li, J. Shi; Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 53 (01, [19] J.-L. Lions; On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp [0] H. Liu, H. Chen; Multiple solutions for an indefinite Kirchhoff-type equation with signchanging potential, Electron. J. Differential Equations, Vol. 015 (015, No. 74, pp [1] Z. Liu, S. Guo; Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys. 66 (015, [] G. Molica Bisci, V. D. Rǎdulescu; Mountain pass theorem for nonlocal equations, Ann. Acad. Sci. Fennicae, 39 (014, [3] B. Ricceri; A general variational principle and some of its applications. J. Comput. Appl. Math. 113 ( [4] B. Ricceri; A further refinement of a three critical points theorem, Nonlinear Anal. 74 (011, [5] B. Ricceri; On an elliptic Kirchhoff-type problem depending on two parameters, J. Glob. Optim. 46 (010, [6] J. Sun, Y. Cheng, T. F. Wu; On the indefinite Kirchhoff type equations with local sublinearity and linearity, Appl. Anal. 016, [7] J. Sun, T. F. Wu; Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 146 (016, [8] J. Sun, T. F. Wu; Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 56 (014, [9] Z. Zhang, K. Perera; Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (006, Addendum posted on September 9, 016 The authors would like to correct Theorems 3.1, 3.3 and their proofs, since the assumptions of Theorems 3.1 and 3.3 can not be verified. Also a new example is given to illustrate Theorem 3.3 instead of the one in Remark 3.4. However, only the case of N = 1 is done and the case of N has not been solved yet. In Theorem 3.1, we assume that N = 1 and = (a, b. The assumption (A3 is removed and the assumption (A4 is replaced by the following. (A5 there exist x 0 (a, b and two constants τ > σ > 0 with B(x 0, σ B(x 0, τ (a, b such that F (x, u 0, for all (x, u (a, b [0, +,

11 EJDE-016/4 INFINITELY MANY SOLUTIONS 11 where α := lim inf t + F (x, u = u 0 α < m(τ σ m 1 (b a β, f(x, sds, for all (x, u (a, b R, F (x, tdx t. b a max ξ t F (x, ξdx t, β := lim t + Moreover, the range of the parameter λ becomes ( m 1 these changes, we restate Theorem 3.1 as follows. (τ σβ, m (b aα. Based on Theorem 4.1. Let N = 1 and = (a, b. Assume that conditions (A1, (A, (A5 m hold. Then for every interval ( 1 m (τ σβ, (b aα, Equation (1.1 has a sequence of solutions {u n } in H0 1 (a, b satisfying Proof. Now we fix λ ( m 1 leads to (τ σβ, lim u n = +. Φ 1 (, r = {u H0 1 (a, b : Φ(u < r} { r } u H0 1 (a, b : u < m { u H0 1 (a, b : u(x 1 r(b a m and Ψ(u = m (b aα. Then for any r > 0, inequality (3.3 b a u 1 b a q r(b a m F (x, udx F (x, udx. } for all x (a, b, Thus, ( v Φ 1 (,r Ψ(v Ψ(u ϕ(r = inf r Φ(u Ψ(u (4.1 r b a r F (x, udx u κ m <. r Let {t n } be a sequence of positive numbers such that t n + and b a u t n F (x, udx = α. (4. lim t n We now choose another positive numbers sequence {r n } with r n = m (b a t n for every n N. It follows from (4.1 and (4. that γ lim inf ϕ(r n (b a m α < +.

12 1 J. SUN, Y. JI, T.-F. WU EJDE-016/4 Since 1 λ < (τ σβ /m 1, there exists a sequence {η n } of positive numbers and µ > 0 such that η n and 1 (τ σ B(x < µ < F (x, η 0,σ ndx λ m 1 ηn. Define v n (x as in (3.5 with = (a, b and it is easy to verify that v n = η n τ σ when N = 1. Moreover, by (A5 one has b Ψ(v n = F (x, v n dx F (x, η n dx. (4.3 a Thus, it follows from (4.3 and (A that for every n N large enough, b I λ (v n = 1 K ( v n λ F (x, v n dx a m 1 v n + m λ F (x, η n dx = m 1ηn τ σ + m λ F (x, η n dx < m 1η n (τ σ (1 λµ + m, which implies that the functional I λ is unbounded from below, since η n + and 1 λµ < 0. It follows that I λ has no global minimum. Therefore, by Theorem.1 (b, there exists a sequence {u n } of critical points of I λ such that The conclusion is achieved. lim u n = +. In Theorem 3.3, we likewise assume that N = 1 and = (a, b. The assumption (A3 is removed and the assumption (A4 is replaced by the following. (A6 there exist x 0 (a, b and three constants ε > 0 and τ > σ > 0 with B(x 0, σ B(x 0, τ (a, b such that where α 0 := lim inf t 0 + F (x, u 0, α 0 < for all (x, u (a, b [0, ε, m(τ σ (b al β 0, b a max ξ t F (x, ξdx B(x t, β 0 := lim 0,σ F (x, tdx t 0 t +. In addition, the range of the parameter λ becomes ( l these, we restate Theorem 3.3 as follows. (τ σβ 0, m (b aα 0. In view of Theorem 4.. Let N = 1 and = (a, b. Assume that conditions (A1, (A, (A6 hold. Then for every interval ( l m (τ σβ, (b aα, Equation (1.1 has a sequence of solutions, which converges strongly to zero in H0 1 (a, b. The proof is omitted here, since it is similar to that of Theorem 3.1. Finally, we give a new example to replace the one in Remark 3.4.

13 EJDE-016/4 INFINITELY MANY SOLUTIONS 13 Example 4.3. Let K(t = 1 + t. Then it is easy to verify that (A1 and (A hold if we choose m = l 1 = l = 1 and q =. Let (a, b = (0, 1, x 0 = 1, τ = 1 3 and σ = 1 4. Then B(x 0, σ B(x 0, τ (0, 1. We take { u(a sin(ln u cos(ln u, if u 0, f(x, u = f(u = 0, if u = 0, where 1 < a < 7 5. A direct calculation shows that { u u (a sin(ln u, if u 0, F (u = f(tdt = 0, if u = 0. It is clear that F (u 0 for all u R, and 0 α 0 = lim inf t 0 + β 0 = lim t 0 + max u t F (u t = a 1 F (t t = a + 1. Moreover, α 0 < m(τ σ (b al β 0, which implies that (A6 holds. Therefore, for every λ ( 1 a+1, a 1, the following problem ( u dx u = λf(u, in (0, 1, 0 u(0 = u(1 = 0, admits infinitely many nontrivial solutions strongly converging at 0 in H 1 0 (0, 1. Acknowledgements. The authors would like to thank Prof. Biagio Ricceri for pointing out these errors and for giving us valuable suggestions during the change process. End of addendum. Juntao Sun (corresponding author College of Science, Hohai University, Nanjing 10098, China, School of Science, Shandong University of Technology, Zibo 55049, China address: sunjuntao008@163.com Yongbao Ji Institute of Finance and Economics, Shanghai University of Finance and Economics, Shanghai 00433, China address: jiyongbao@16.com Tsung-fang Wu Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan address: tfwu@nuk.edu.tw