INFINITELY MANY SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS

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1 Electronic Journal of Differential Equations, Vol , No. 188, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu INFINITELY MANY SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS WEI-BING WANG, WEI TANG Abstract. In this article, we study a Kirchhoff tye roblem with nonlinear Neumann boundary conditions on a bounded domain. By using variational methods, we rove the existence of infinitely many solutions. 1. Introduction In this work, we study the multilicity of solutions for the ellitic roblem [ 1 M u dx] u = fx, u, in, u u = λku + µgu, on, ν where Mt = a + bt, > N, a > 0, b 0, is a nonemty bounded oen subset of R N with a boundary of class C 1, u ν is the outer unit normal derivative, u := div u 2 u is the -Lalacian oerator, λ, µ are ositive real arameters, the functions f, k, g satisfy hyotheses stated as follows: H1 k, g CR, R and there exist two ositive constants ρ, ρ such that Ku + Gu ρ u ρ + 1 for all u R, where Ku = u 0 ksds, Gu = u 0 gsds. H2 f C R, R and there exist two ositive constants a 1, a 2 such that a 1 u F x, u a 2 u for all x, u R, where F x, u = u fx, sds. 0 H3 f C R, R and there exist two ositive constants ϱ, ϱ such that F x, u ϱ u ϱ + 1 for all x, u R. H4 there exist two ositive constants b 1, b 2 such that for all u R. b 1 u Gu b 2 u 2010 Mathematics Subject Classification. 35J60, 35J20. Key words and hrases. Kirchhoff tye equation; weak solution; critical oint. c 2016 Texas State University. Submitted December 18, Published July 13,

2 2 W.-B. WANG, W. TANG EJDE-2016/188 Problem 1.1 is the nonlocal roblem, which is related to the model introduced by Kirchhoff [15], ρ 2 u t 2 ρ0 h + E L u 2 u 2L x 2 dx = 0, 1.2 x2 0 which extends the classical D Alembert s wave equation by considering the effects of the changes in the length of the strings during the vibrations. Interest of the mathematicians on the nonlocal roblems has increased because they reresent a variety of relevant hysical and engineering situations[4, 12]. Many interesting results for Kirchhoff tye roblems were obtained and we refer to[1, 2, 3, 8, 9, 11, 13, 14, 17, 18, 19] and references therein for an overview on these subjects. Relatively seaking, Kirchhoff tye roblems with nonlinear boundary conditions have rarely been considered. In addition, when involving the existence of infinitely many solutions, most results assume that nonlinear term is odd in order to aly some variant of the classical Lusternik-Schnirelmann theory and only a few aers deal with nonlinearities having no symmetry roerties[5, 6, 16]. The main urose of this article is to establish the existence of infinitely many solutions for 1.1 without the assumtion of symmetry roerty, by adoting the framework of Bonanno and Molica Bisci [6]. 2. Preliminaries Let X be a reflexive real Banach sace and I λ : X R a functional satisfying the structure hyothesis: H5 I λ u = Ψu λφu for all u X, where Ψ, Φ : X R are two functions of class C 1 on X with Ψ coercive, i.e. lim u + Ψu = +, and λ is a real arameter. Provided that inf X Ψ < r, ut φ Iλ r := inf u Ψ 1 ],r[ suu Ψ 1 ],r[ Φu Φu, r Ψu γ := lim inf r + φ I λ r, δ := lim inf r inf X Ψ + φ Iλ r. When γ = 0 or δ = 0 we agree to read 1 γ or 1 δ as +. Our main tool is a smooth version of critical oint theorem which are recalled below, see [6]. Theorem 2.1. Assume that H5 holds. Then: a For each r > inf X Ψ and every λ ]0, 1/φ Iλ r[, the restriction of the functional I λ to Ψ 1 ], r[ has a global minimum, which a s critical oint local minimum of I λ in X. b If γ < +, then, for each λ ]0, 1/γ[, the following alternative holds:either 1 I λ ossess a global minimum, or 2 there is a sequence {u n } of critical oints of I λ such that lim n Ψu n = +. c If δ < +, then, for each λ ]0, 1/δ[, the following alternative holds: either 1 there is a global minimum of Φ which is a local minimum of I λ, or 2 there is a sequence {u n } of airwise distinct critical oints of I λ such that lim n Ψu n = inf X Ψ, which weakly converges to a global minimum of Φ.

3 EJDE-2016/188 INFINITELY MANY SOLUTIONS 3 or Let W 1, be the usual Sobolev sace endowed with the norm 1/ u := u + u dx u := u dx + u dσ 1/, where dσ is the measure on the boundary. Clearly, is equivalent to. Let k := su u W 1, \{0} max x ux max, k := su x ux 2.1 u u W 1, \{0} u Since > N, the embedding W 1, C is comact, and thus 0 < k, k <, ux k u, ux k u for u W 1,. 2.2 A weak solution of roblem 1.1, we mean that a function u W 1, satisfies [ ] 1 M u dx u 2 u vdx fx, uvdx λku + µguvdσ = 0 for every v W 1,. Define the functionals on W 1, by R u dx { 1 Γu = 1 [ M 1 b a + b sds = 2 u dx ] a, b > 0, a 1 0 u dx, b = 0, µ ψu = Γu F x, udx, ϕu = Ku + λ Gu dσ, ψ u = Γu µ Gudσ, ϕ u = Kudσ + 1 F x, udx, λ J λ u = ψu λϕu, I λ u = ψ u λϕ u. Conditions H1 and H2 or H1 and H3 and > N guarantee that ψ, ϕ or ψ, ϕ are well defined and of class C 1. Moreover, J λu, v = I λu, v [ ] 1 = M u dx u 2 u vdx fx, uvdx λku + µguvdσ. Hence, the critical oints of J λ or I λ are the weak solutions of Main results Put = { a 1, a } 1, b3 = { a 1, b } 1, = Moreover, let dx, = max t ξ Kt max t ξ Kt A := lim inf ξ + ξ, A 0 := lim inf ξ 0 + ξ, dσ.

4 4 W.-B. WANG, W. TANG EJDE-2016/188 G := lim su F := lim su ξ + B := lim su ξ + ξ + Kξ ξ, B Kξ 0 := lim su ξ 0 ξ +, max t ξ Gt max t ξ Gt ξ, G 0 := lim su ξ 0 ξ +. max t ξ F x, tdx ξ, F 0 := lim su max t ξ F x, tdx ξ 0 ξ + We resent our main results as follows. Theorem 3.1. Assume that H1, H2 hold and there exist two real sequences {α n }, {β n } with lim n β n = + and a ositive constant ρ > 0 such that α n < 1 a3 1/βn, Gt 0, t ρ, k a 2 max t βn Kt Kα n A := lim n βn k a 2 a 1 < B 3 α n k, a 2 max t βn Gt Gα n G := lim n βn k a 2 a 1 < +. 3 α n Then for each λ Λ :=]λ 1, λ 2 [, where there exists µ λ > 0, where λ 1 = a 2, λ 2 = B k, A µ λ = 1 a3 G k λa, such that for all µ [0, µ λ [, 1.1 has an unbounded sequence of weak solutions. Proof. First, we observe that owing to the condition A < B /k a 2, the interval Λ is non-emty. Moreover, for each fixed λ Λ and taking into account that A λ < a3 /k, one has 0 < µ λ <. Our aim is to aly Theorem 2.1. For this end, we show γ < +, where γ is defined in Theorem 2.1. By Assumtion H2, we have u ψu max { 2a 1 Put r n = β n /k for all n N, by 2.2 and 3.1, one has Hence, φ Jλ r n = }, a 2 u + 2b 1 u ψ 1 ], r n ] {u W 1, : u β n }, ψα n = F x, α n dx a 2 αn < r n. inf u ψ 1 ],r n[ suu ψ 1 ],r n[ ϕu ϕu r n ψu max t βn [Kt + µ λ Gt] ϕu inf ψu<r n r n ψu max t β n [Kt + µ λ Gt] ϕα n r n ψα n

5 EJDE-2016/188 INFINITELY MANY SOLUTIONS 5 max t β n [Kt Kα n ] r n a 2 αn k A + µ λ G < +. + µ λ Since µ [0, µ λ[, γ lim inf φ J n + λ r n < k A + µ λ λ G max t βn [Gt Gα n ] r n a 2 α n = 1 λ < + ; that is, 0 < λ 1 < λ < 1/γ. The condition b of Theorem 2.1 can be alied and either J λ has a global minimum or there exists a sequence {u n } of weak solutions of the roblem 1.1 such that u n as n. Now, we verify that J λ is unbounded from below. First, assume that B = +. Accordingly, fixed C with C > a 2 / λ and {c n } be a sequence of ositive numbers with c n + as n such that Kc n > Cc n, n sufficiently large. Taking the sequence {v n } W 1, such that v n x = c n, x, for the sufficiently large n, one has J λv n = F x, v n dx λ Kv n dσ µ Gv n dσ a 2 c n λ Kv n dσ a 2 C λ c n; that is, J λ as n. Next, assume that B < +. Since λ > λ 1 = a 2 / B, we fix 0 < ε < B λ a2. Let {c n} be a sequence of ositive numbers with c n + as n such that B εc n < Kc n < B + εc n, n sufficiently large Arguing as before and by choosing v n c n, for the sufficiently large n, one has J λv n = F x, v n dx λ Kv n dσ µ Gv n dσ a 2 c n B ε λ c n as n. Hence, J λ is unbounded from blew and the roof is comlete. Corollary 3.2. Assume that H1, H2 hold. Further suose that G < +, A < B /k a 2 and there exists ρ > 0 such that Gt 0 for all t ρ. Then for each λ ]λ 3, λ 4 [, where there exists µ λ > 0, where λ 3 = a 2, λ 4 = B k A, µ λ = 1 a3 G k λa, such that for all µ [0, µ λ [, 1.1 has an unbounded sequence of weak solutions.

6 6 W.-B. WANG, W. TANG EJDE-2016/188 Proof. Let {β n } be a sequence of ositive numbers which aroaches infinity such that max t βn Kt A =. lim ξ + Taking α n = 0 for every n N and noting that max t βn Gt G = lim n + βn G, from Theorem 3.1, we obtain the conclusion. Alying art c of Theorem 2.1, we get the following theorem. Theorem 3.3. Assume that H1, H2 hold and there exist two real sequences {α n }, {β n } with lim n β n = 0 and ositive constant ρ > 0 such that α n < 1 a3 1/βn, Gt 0, 0 t ρ, k a 2 max t βn Kt Kα n A 0 := lim n βn k a 2 a 1 < B 0 3 α n k, a 2 max t βn Gt Gα n G 0 := lim n βn k a 2 a 1 < +. 3 α n Then for each λ ]λ 4, λ 5 [, where there exists µ λ > 0, where β n λ 4 = a 2, λ 5 = B 0 k, A 0 µ λ = 1 a3 G 0 k λa 0, such that for all µ [0, µ λ [, 1.1 has a sequence of weak solutions, which converges strongly to zero. Corollary 3.4. Assume that H1, H2 hold. Further suose that G 0 < +, A 0 < B 0 /k a 2 and there exists ρ > 0 such that Gt 0 for all 0 < t ρ. Then for each λ ]λ 7, λ 8 [, where there exists ˆµ λ > 0, where λ 7 = a 2, λ 8 = B 0 k A 0, ˆµ λ = 1 a3 G 0 k λa 0, such that for all µ [0, ˆµ λ [, 1.1 has a sequence of weak solutions, which converges strongly to zero. Now, we consider the case when H1, H3, H4 hold. Theorem 3.5. Assume that H1, H3, H4 hold and F x, u 0 for x, u r > 0. If there exists the ositive constant λ such that λb > b 2, F < b 3 k λ A, then 1.1 has an unbounded sequence of weak solutions for λ = λ, µ = 1.

7 EJDE-2016/188 INFINITELY MANY SOLUTIONS 7 Proof. Let λ = λ, µ = 1 and {β n } be a sequence of ositive numbers with β n + as n such that max t βn Kt A = lim n + βn. By Assumtion H4, we have b 3 u ψ u max { 2a 1 }, b 2 u + 2b 1 Put r n = β nb 3 /k for all n N, by 2.2 and 3.2, one has Hence, φ Iλ r n = ψ 1 ], r n ] {u W 1, : u β n }. u ψ 1 inf ],r n[ su u ψ 1 ],r ϕ n[ u r n max {u X: u β n} ϕ u r n k max t βn Kt b 3 βn + 1 λ suu ψ 1 ],r ϕ n[ u ϕ u r n ψ u max t β n F x, tdx β n k b 3 A + 1 λf < 1 λ. The rest roof is similar to that of Theorem 3.1 and we omit it. u Theorem 3.6. Assume that H1, H3, H4 hold and F x, u 0 for x, 0 u rr > 0. If there exists the ositive constant λ such that λb 0 > b 2, F 0 < b 3 k λ A 0, then 1.1 has a sequence of weak solutions for λ = λ, µ = 1, which converges strongly to zero. 4. Examles In this section, we resent the two examles which rovide the roblems that admit infinitely many solutions. Examle 4.1. Consider the differential equation [ 1 ] 1 + b u 2 dx u x + uu cos u + 2 sin u + 4 = 0, x 0, 1, where b 0, Then 0 u 0 = λku0 + µgu0, ku = { u 2 sinln u, u > 0, 0, u 0, fu = uu cos u + 2 sin u + 4, u 1 = λku1 + µgu1, gu = u sin u. F u = u sin u, 4.1

8 8 W.-B. WANG, W. TANG EJDE-2016/188 Ku = { u 3 8 [3 sinln u cosln u], u > 0, 0, u 0, Gu = 1 2 u2 + cos u 1, = = 1, a 1 = 1, a 2 = 3, = 1 2. According to [7, Remark 1], one has the estimate k 2. Taking α n = e 2n+1π, β n = e 2n+1π, we easily obtain that A = 0, B = +, G = e2π 1 2e 2π 6k 2. By Theorem 3.1, 4.1 has an unbounded sequence of weak solutions for λ > 0, 0 µ < e2π 6k 2 k 2 e 2π 1. Examle 4.2. Consider the differential equation [ 1 M u dx] u = cx u ρ 1 u, in, 2 u u ν = λku u 2 u, on, 4.2 where Mt = a + bt, > N, a > 0, b 0, 1 < ρ <, c C and cx 0, is a nonemty bounded oen subset of R N with a boundary of class C 1, k 1 = 2, k n+1 = kn 12, l n = kn 10, n N, ln 0.5 kn l n t, l n 1 t l n + 1, n 1, kt = k +0.5 n+1 ln k n+1 t, k n+1 1 t k n+1 + 1, n 1, 0, otherwise. Noting that Kk n + 1 = k +0.5 n k , Kl n + 1 = ln 0.5 k , n 2, we have Kk n + 1 Kl n + 1 lim = +, lim n k n + 1 n l n + 1 = 0. Hence, A = 0, B = +. It is easy to check that F = 0. By Theorem 3.5, 4.2 has an unbounded sequence of weak solutions for all λ > 0. Acknowledgments. The authors exress their gratitude to the reviewers for careful reading and helful suggestions which led to an imrovement of the original manuscrit. The work is suorted by Hunan Provincial Natural Science Foundation of China 2015JJ2068 and NNSF of China References [1] C. O. Alves, F. J. S. A. Corrêa, T. F. Ma; Positive solutions for a quasilinear ellitic equation of Kirchhoff tye, Comut. Math. Al., , no.1, [2] G. Anello; A uniqueness result for a nonlocal equation of Kirchhoff tye and some related oen roblem, J. Math. Anal. Al., , no. 1, [3] G. Autuori, P. Pucci, M. C. Salvatori; Asymtotic Stability for anisotroic Kirchhoff system, J. Math. Anal. Al., , no. 1, [4] G. Molica Bisci; Variational roblems on the Shere, Recent Trends in Nonlinear Partial Differential Equations II. Stationary roblems, Contem. Math., ,

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