MULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS INVOLVING CONCAVE AND CONVEX NONLINEARITIES IN R 3

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1 Electronic Journal of Differential Equations, Vol (2016), No. 301, ISSN: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu MULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS INVOLVING CONCAVE AND CONVEX NONLINEARITIES IN XIAOFEI CAO, JUNXIANG XU, JUN WANG Abstract. In this article, we consider the multilicity of ositive solutions for a class of Kirchhoff tye roblems with concave and convex nonlinearities. Under aroriate assumtions, we rove that the roblem has at least two ositive solutions, moreover, one of which is a ositive ground state solution. Our aroach is mainly based on the Nehari manifold, Ekeland variational rincile and the theory of Lagrange multiliers. 1. Introduction and statement of main results In this article, we consider the multilicity of ositive solutions for the Kirchhoff tye roblem (a + b u 2 dx) u + V (x)u = f(x) u q 2 u + g(x) u 2 u, u H 1 ( ), (1.1) where a and b are ositive constants, 1 < q < 2, 4 < < 2 = 6, V (x), f(x), g(x) are continuous functions and satisfy suitable conditions. Problem (1.1) can be written in the general form (a + b u 2 dx) u + V (x)u = h(x, u), R N u H 1 (R N ), (1.2) where V : R N R is a continuous otential, a, b > 0 are constants. For the case of the nonlinearity h is asymtotically linear or suerlinear, it has been studied extensively by many authors, see [12, 13, 17, 18, 19, 20, 24, 26, 31, 38, 39] and their references therein. A secial case of (1.2) is the well-known equation (a + b u 2 dx) u = h(x, u) in Ω, Ω (1.3) u = 0 on Ω, 2010 Mathematics Subject Classification. 35A01, 35A15. Key words and hrases. Kirchhoff tye roblem; Ekeland variational rincile; Nehari manifold. c 2016 Texas State University. Submitted July 25, Published November 25,

2 2 X. CAO, J. XU, J. WANG EJDE-2016/301 where Ω is a bounded domain in R N. This roblem (1.3) is related to the stationary analogue of the equation u tt (a + b u 2 dx) u = h(x, u) in Ω, Ω (1.4) u = 0 on Ω, roosed by Kirchhoff [21] in 1883 as an extension of the classical D Alembert s wave equation for free vibration of elastic strings. Kirchhoff s model takes into account the changes in length of the string roduced by transverse vibrations. In (1.4), u denotes the dislacement, h(x, u) the external force and b the initial tension while a is related to the intrinsic roerties of the string (such as Young s modulus). Such roblems are often viewed as nonlocal because of the resence of the integral term Ω u 2 dx which imlies that roblem (1.4) is no longer a ointwise identity. This henomenon causes some mathematical difficulties which makes the study of such a class of roblem articularly interesting. Besides, similar nonlocal roblem also aears in other fields such as hysical and biological systems, where u describes a rocess that deends on the average of itself, for examle, the oulation density. Here we are interested in the nonlinearity h made u of the combination of a sublinear term and a suerlinear term. This case was considered by Willem [35] for the ellitic equation u = λ u q 2 u + µ u 2 u, (1.5) u H0 1 (Ω), where 1 < q < 2 < < 2 (2 = 2N/(N 2) if N 3, 2 = if N = 1, 2) and Ω is a bounded domain in R N. The author roved that for every µ > 0, λ R, roblem (1.5) has a sequence of high energy solutions, and for every λ > 0, µ R, roblem (1.5) has a sequence of negative energy solutions. Ambrosetti, Brezis and Cerami [1] studied equation (1.5) when µ = 1. The authors used sub- and suer-solutions to rove that when µ = 1 there exists Λ > 0 such that (1.5) admits at least two ositive solutions for λ (0, Λ), one ositive solution for λ = Λ and no ositive solution for λ > Λ. Note that if µ 0 in (1.5), then by scaling (1.5) becomes the situation of µ = 1. Wu [36] also studied the concave-convex ellitic equation u + u = f λ (x)u q 1 + g µ (x)u 1 in R N, u 0 in R N, u H 1 (R N ) where 1 < q < 2 < < 2 (2 = 2N/(N 2) if N 3, 2 = if N = 1, 2), f λ (x) = λf + (x) + f (x)(f + (x) := + max{0, +f(x)} 0) (1.6) is sign-changing, g µ (x) = a(x)+µb(x) and the arameters λ, µ 0. When the functions f + (x), f (x), a(x), b(x) satisfy suitable conditions, he roved the multilicity of ositive solutions for the roblem (1.6). Recently, Chen, Kuo and Wu [9] considered the Kirchhoff tye roblem (a + b u 2 dx) u = λf(x) u q 2 u + g(x) u 2 u, Ω (1.7) u H0 1 (Ω),

3 EJDE-2016/301 KIRCHHOFF TYPE PROBLEMS 3 where Ω is a smooth bounded domain in R N with 1 < q < 2 < < 2 (2 = 2N/(N 2) if N 3, 2 = if N = 1, 2), the arameters a, b, λ > 0 and the weight functions f, g C(Ω). Using Nehari manifold they roved the existence of solutions for (1.7) with > 4, = 4 and < 4, resectively. Motivated by these aers [9, 35, 36], we consider the Kirchhoff roblem (1.1) with otential V (x) and concave-convex nonlinearities on the whole sace. To the best of our knowledge, there are few aers which deal with this tye of Kirchhoff roblem (1.1). The main difficulties lie in the unboundedness of the domain, the resence of the nonlocal term and the concave-convex nonlinearities. Assume that V (x), f(x), g(x) satisfy the following conditions: (A1) V (x) C(, R), V 0 := inf V (x) > 0 and for any M > 0, there exists a constant r 0 > 0 such that meas({x B r0 (y) : V (x) M}) 0 as y +, where B r0 (y) denotes the ball centered at y with radius r 0, meas denotes the Lebesgue measure in. (A2) f C( ) L q ( ), where q = /( q). (A3) g C( ) L ( ) and g(x) > 0, for almost every x. Let σ := ( 2)(2 q) (2 q)/( 2) ( S q )( q)/( 2) and 0 < σ := q 2σ < σ, where S is the best Sobolev constant described in the following Lemma 2.2. Theorem 1.1. Under the assumtions (A1) (A3), if f q g (2 q)/( 2) (0, σ), the roblem (1.1) has at least two ositive solutions, one of which has negative (0, σ ), the solution corresonding to the negative energy is a ositive ground state solution and the other one corresonds to ositive energy. energy. In articular, if f q g (2 q)/( 2) In roblem (1.1), because of the unboundedness of the domain there is no comactness, thus we bring in the hyothesis (A1) to recover the comactness. Moreover, the resence of the nonlocal term and the concave-convex nonlinearities revents us from using the Nehari manifold method in a standard way as [12, 17, 18, 19, 20, 24, 26, 31, 38, 39]. Motivated by aers [6, 9, 36], we connect the Nehari manifold with the fibering ma and slit the Nehari manifold into three arts which are then considered searately. By utting suitable conditions on continuous to a suitable range, we use Ekeland variational rincile and the theory of Lagrange multiliers to obtain two ositive solutions of the roblem (1.1). In addition, from the condition (A2), we easily see that f(x) is allowed to be sign-changing as [36]. functions f(x), g(x) and restricting f q g (2 q)/( 2) Remark 1.2. The condition (A1) was first introduced by Bartsch and Wang in [5]. Note that it is weaker than both conditions (1) V (x) C(, R), inf V (x) > 0, V (x) + as x + (See [19]). (2) V (x) C(, R), inf V (x) > 0, for each M > 0, meas({x : V (x) M}) < (See [10, 37, 40]). These conditions are often used to recover comactness. Remark 1.3. We can also consider the Kirchhoff roblem (a + b ( u 2 + V (x)u 2 ) dx)( u + V (x)u) = f(x) u q 2 u + g(x) u 2 u, R N u H 1 (R N ), (1.8)

4 4 X. CAO, J. XU, J. WANG EJDE-2016/301 where the arameters a, b > 0 and 1 < q < 2 < < 2 (2 = 2N/(N 2) if N 3, 2 = if N = 1, 2). Under the assumtions of Theorem 1.1, restricting f q g (2 q)/( 2) to a suitable range as in Theorem 1.1, we can also obtain the existence of solutions for (1.8). It is worth noting that (1.8) is similar to (1.7) in the case of the whole sace R N and that there is comactness because of condition (A1). Hence we can also obtain the results in [9]. However, this tye of Kirchhoff (1.8) is different from that of Kirchhoff (1.1). This article is organized as follows: Section 2 is dedicated to our abstract framework and some reliminary results. Section 3 is concerned with the roof of Theorem 1.1. Throughout this aer, C and C i are used in various laces to denote distinct constants. L (R N ) is the usual Lebesgue sace endowed with the standard norm u = ( R N u dx) 1/ for 1 < and u = su x R N u(x) for =. When it causes no confusion, we still denote by {u n } a subsequence of the original sequence {u n }. 2. Preliminary results In this section, we recall some reliminaries and establish the variational setting for our roblem. Under the assumtion (A1), define E := {u H 1 ( ) : V (x)u 2 dx < + }, with the associated norm ( ) 1/2, u = (a u 2 + V (x)u 2 ) dx where H 1 ( ) is the well known Sobolev sace. Then the energy functional corresonding to (1.1) is I(u) = 1 (a u 2 + V (x)u 2 ) dx + b ( ) 2 u 2 1 dx f(x) u q dx 2 R 4 3 R q 3 1 (2.1) g(x) u dx, u E. Lemma 2.1. If (A1) (A3) hold, then the functional I C 1 (E, R) and for any u, v E, ( ) I (u), v = a + b u 2 dx u v dx + V (x)uv dx R 3 (2.2) f(x) u q 2 uv dx g(x) u 2 uv dx. The roof of the above lemma is a direct comutation, and can be found in [35, 37]. As ointed out reviously, assumtion (A1) is used to recover comactness of embedding theorem, which is given in the next lemma. Lemma 2.2 ([5, 40]). Under assumtion (A1), the embedding E L ( ) is continuous for [2, 2 ] and comact for [2, 2 ). Throughout this aer, we denote by S the best Sobolev constant for the embedding E L ( ) which is given by S = inf u E\{0} u 2 ( > 0. u R dx) 2/ 3

5 EJDE-2016/301 KIRCHHOFF TYPE PROBLEMS 5 In articular, u S 1/2 u, u E\{0}. It is well-known that finding a weak solution of roblem (1.1) is equivalent to finding a critical oint of the corresonding functional I. In the following, we are devoted to finding the critical oint of the corresonding functional I. As usual, some energy functional such as I in (2.1) is not bounded from below on E but, as we will see, is bounded from below on an aroriate subset of E and a minimizer on this set (if it exists) may give rise to a solution of corresonding differential equation. A good exemlification for an aroriate subset of E is the so-called Nehari manifold N = {u E : I (u), u = 0}, where, denotes the usual duality between E and E. It is clear to see that u N if and only if u 2 + b u 4 2 = f(x) u q dx + g(x) u dx. (2.3) Obviously, N contains all solutions of (1.1). Below, we shall use the Nehari manifold methods to find critical oints for functional I. The Nehari manifold N is closely linked to the behavior of functions of the form K u : t I(tu) for t > 0. Such mas are known as fibering mas, which were introduced by Drábek and Pohozaev in [15]. For u E, let K u (t) = I(tu) = 1 2 t2 u 2 + b 4 t4 u q tq f(x) u q dx 1 R t g(x) u dx; 3 R 3 K u(t) = t u 2 + t 3 b u 4 2 t q 1 f(x) u q dx t 1 g(x) u dx; K u(t) = u 2 + 3t 2 b u 4 2 (q 1)t q 2 f(x) u q dx ( 1)t 2 g(x) u dx. Lemma 2.3. Let u E\{0} and t > 0. Then tu N if and only if K u(t) = 0, that is, the critical oints of K u (t) corresond to the oints on the Nehari manifold. In articular, u N if and only if K u(1) = 0. Proof. The result is an immediate consequence of the fact that K u(t) = I (tu), u = 1 t I (tu), tu. Thus, it is natural to slit N into three arts corresonding to local minima, oints of inflection and local maxima. Accordingly, we define N + = {u N : K u(1) > 0}, N 0 = {u N : K u(1) = 0}, N = {u N : K u(1) < 0}. It is easy to see that K u(1) = u 2 + 3b u 4 2 (q 1) f(x) u q dx ( 1) g(x) u dx. (2.4)

6 6 X. CAO, J. XU, J. WANG EJDE-2016/301 Define Ψ(u) = K u(1) = I (u), u = u 2 + b u 4 2 f(x) u q dx g(x) u (2.5) dx. Then for u N, d dt Ψ(tu) t=1 = Ψ (u), u = Ψ (u), u I (u), u = K u(1) = u 2 + 3b u 4 2 (q 1) f(x) u q dx ( 1) g(x) u dx. For each u N, Ψ(u) = K u(1) = 0. Thus, we have K u(1) = K u(1) (q 1)Ψ(u) = (2 q) u 2 + (4 q)b u 4 2 ( q) g(x) u (2.6) dx and K u(1) = K u(1) ( 1)Ψ(u) = (2 ) u 2 + (4 )b u ( q) f(x) u q (2.7) dx. To ensure the Nehari manifold N to be a C 1 -manifold, we need the following roosition. Let σ := ( 2)(2 q) (2 q)/( 2) ( S q )( q)/( 2). Proosition 2.4. If f q g (2 q)/( 2) (0, σ), then the set N 0 = {0}. Proof. Suose, on the contrary, there exists u N \{0} such that K u(1) = 0. By Lemma 2.2, g(x) u dx g S /2 u. (2.8) Noting that 1 < q < 2 and 4 < < 6, from (2.6) we have (2 q) u 2 ( q) g S /2 u, and so ( (2 q)s 2 ) 1 2 u. (2.9) ( q) g Moreover, by Hölder inequality and Lemma 2.2, we have ( ) 1/q ( ) q/ f(x) u q dx f(x) q dx u dx (2.10) = f q u q f q S q/2 u q. From (2.7) we have ( 2) u 2 ( q) f q S q/2 u q, which imlies that ( ( q) f q ) 1 2 q u. (2.11) ( 2)S q 2 Combining (2.9) and (2.11) we deduce that 2 q 2 f q g ( (2 q)s 2 ) 2 q 2 2 q q S q 2 = ( 2)(2 q) 2 q 2 ( S q ) q 2,

7 EJDE-2016/301 KIRCHHOFF TYPE PROBLEMS 7 which contradicts the assumtions. The roof is comlete. Let h b (t) := t 2 q u 2 + t 4 q b u 4 2 t q g(x) u dx, then we have K u(t) = t q 1( ) h b (t) f(x) u q dx. (2.12) Clearly, h b (0) = 0, h b (t) > 0 for t is small enough and h b (t) as t. From 1 < q < 2, 4 < < 2 = 6 and h b(t) = t q 1( ) (2 q)t 2 u 2 + (4 q)t 4 b u 4 2 ( q) g(x) u dx = 0, we can infer that there is a unique t b,max > 0 such that h b (t) achieves its maximum at t b,max, increasing for t [0, t b,max ) and decreasing for t (t b,max, ) with lim t h b (t) =. Proosition 2.5. Suose that f q g (2 q)/( 2) (0, σ) and u E\{0}. Then (i) if f(x) u q dx 0, then there is a unique t > t b,max such that t u N and I(t u) = su I(tu); t 0 (ii) if f(x) u q dx > 0, then there are unique t + and t with 0 < t + < t b,max < t such that t + u N +, t u N and I(t + u) = Proof. Since b > 0, we have inf I(tu), I(t u) = su I(tu). 0 t t b,max t t b,max h b (t) > h 0 (t) = t 2 q u 2 t q g(x) u dx, where h 0 (t) = h b (t) b=0. It is clear that h 0 (t) has a unique critical oint at t 0,max = t 0,max (u), where ( (2 q) u 2 ) 1 t 0,max = ( q) 2. g(x) u R dx 3 It follows that h 0 (t 0,max ) = u q( u g(x) u R dx 3 u q( u g S /2 u ) 2 q ( 2 2 q q ) 2 q 2 ( 2 q q = u q( (2 q)s 2 ) 2 q 2 2 g ( q) q > 0. ) 2 q 2 2 q ) 2 q 2 2 q Thus, h b (t b,max ) > h 0 (t 0,max ) > 0. From f q g (2 q)/( 2) (0, σ), (2.10) and (2.13) we also have ( (2 q)s f(x) u q dx < u q 2 ) 2 q 2 2 R g 3 ( q) q h 0 (t 0,max ) < h b (t b,max ). (2.13) (2.14)

8 8 X. CAO, J. XU, J. WANG EJDE-2016/301 (i) If f(x) u q dx 0, noting that h b (0) = 0 and h b (t) as t, by (2.12) there is a unique t > t b,max such that K u(t ) = 0, that is t u N. Moreover, for tu N, K u(t) = 0. By (2.12) we obtain that K u(t) = t q 1 h b(t) < 0. Thus, if f(x) u q dx 0, there is a unique t > t b,max such that t u N and I(t u) = su I(tu). t 0 (ii) If f(x) u q dx > 0, by (2.12) and (2.14) we know there are unique t + and t with 0 < t + < t b,max < t such that K t + u (1) = 0, K t u (1) = 0, that is t + u, t u N. From K u(t) = t q 1 h b (t) and h b (t+ ) > 0 > h b (t ), we have t + u N +, t u N and I(t + u) = inf I(tu), I(t u) = su I(tu). 0 t t b,max t t b,max The forthcoming lemma obtains the minimizing sequence of the energy functional I on Nehari manifold N. Lemma 2.6. The energy functional I is coercive and bounded from below on N. Proof. For u N, then, by Hölder inequality and Lemma 2.2, I(u) = I(u) 1 4 I (u), u = 1 ( 1 4 u 2 q 1 4) ( 1 f(x) u q dx + R 4 1 ) g(x) u dx 3 1 ( 1 4 u 2 q 1 f q S 4) q/2 u q. This comletes the roof. Lemma 2.7. If f q g (2 q)/( 2) (0, σ), the set N is closed in E. Proof. Let {u n } N such that u n u in E. In the following we rove u N. Indeed, by I (u n ), u n = 0 and I (u n ), u n I (u), u = I (u n ) I (u), u + I (u n ), u n u 0, as n, we have I (u), u = 0. So u N. For any u N, from (2.6) we have (2 q) u 2 + (4 q)b u 4 2 < ( q) g(x) u dx. Similar to the roof of (2.9), we have u ( (2 q)s 2 ) 1 2. (2.15) ( q) g Hence N is bounded away from 0. By (2.6), it follows that K u n (1) K u(1). From K u n (1) < 0, we have K u(1) 0. By Proosition 2.4, for f q g (2 q)/( 2) (0, σ), K u(1) < 0. Thus we deduce u N.

9 EJDE-2016/301 KIRCHHOFF TYPE PROBLEMS 9 The following lemma yields a (P S) c sequence from the minimizing sequence of the energy functional I on Nehari manifold N. Lemma 2.8. If f q g (2 q)/( 2) (0, σ), then for every u N +, there exist ɛ > 0 and a differentiable function ϕ + : B ɛ (0) R + := (0, + ) such that and where ϕ + (0) = 1, ϕ + (w)(u w) N +, w B ɛ (0) (ϕ + ) (0), w = L(u, w)/k u(1), (2.16) L(u, w) = 2 u, w + 4b u 2 u w dx q f(x) u q 2 uw dx g(x) u 2 uw dx. Moreover, for any C 1, C 2 > 0, there exists C > 0 such that if C 1 u C 2, (ϕ + ) (0), w C w. Proof. We define F : R E R by F (t, w) = K u w(t), it is easy to see F is differentiable. Since F (1, 0) = 0 and F t (1, 0) = K u(1) > 0, we aly the imlicit function theorem at oint (1, 0) to obtain the existence of ɛ > 0 and differentiable function ϕ + : B ɛ (0) R + := (0, + ) such that ϕ + (0) = 1, F (ϕ + (w), w) = 0, w B ɛ (0). Thus, ϕ + (w)(u w) N, w B ɛ (0). Next, we rove ϕ + (u w) N +, w B ɛ (0). Indeed, by u N + and the set N N 0 is closed, we know dist(u, N N 0 ) > 0. Since ϕ + (w)(u w) is continuous with resect to w, when ɛ is small enough, we know for w B ɛ (0) so ϕ + (w)(u w) u < 1 2 dist(u, N N 0 ), ϕ + (w)(u w) N N 0 dist(u, N N 0 ) dist(ϕ + (w)(u w), u) > 1 2 dist(u, N N 0 ) > 0. Thus, ϕ + (w)(u w) N + for all w B ɛ (0). Also by the differentiability of the imlicit function theorem, we have (ϕ + ) (0), w = F w(1, 0), w. F t (1, 0) Note that L(u, w) = F w (1, 0), w and K u(1) = F t (1, 0). So we rove (2.16). Next we rove that for any C 1, C 2 > 0, if C 1 u C 2, u N +, there exists δ > 0 such that K u(1) δ > 0. On the contrary. If there exists a sequence {u n } N +, C 1 u n C 2, such that for any δ n sufficiently small, K u n (1) δ n, δ n 0 as n. From (2.6) we have (2 q) u n 2 + (4 q)b u n 4 2 = ( q) g(x) u n dx + O(δ n ),

10 10 X. CAO, J. XU, J. WANG EJDE-2016/301 where O(δ n ) 0 as n. Noting that 1 < q < 2, 4 < < 6, C 1 u n C 2 and (2.8), we have (2 q) u n 2 ( q) g S /2 u n + O(δ n ), and so ( (2 q)s 2 ) 1 2 u n + O(δ n ). (2.17) ( q) g From (2.7) we also have ( 2) u n 2 + ( 4)b u n 4 2 = ( q) f(x) u n q dx + O(δ n ). In view of (2.10), we have ( 2) u n 2 ( q) f q S q/2 u n q + O(δ n ), which imlies ( ( q) f q ) 1 2 q u n + O(δ ( 2)S q n ). (2.18) 2 Combining (2.17) and (2.18) as n, we deduce a contradiction. Thus if C 1 u C 2, there exist C > 0 such that This comletes the roof. (ϕ + ) (0), w C w. Similarly, we establish the following lemma. Lemma 2.9. Assume f q g (2 q)/( 2) (0, σ), then for every u N, there exist ɛ > 0 and a differentiable function ϕ : B ɛ (0) R + := (0, + ) such that ϕ (0) = 1, ϕ (w)(u w) N, w B ɛ (0), (ϕ ) (0), w = L(u, w)/k u(1), where L(u, w) is defined in Lemma 2.8. Moreover, for any C 1, C 2 > 0, there exists C > 0 such that if C 1 u C 2, (ϕ ) (0), w C w. The following lemma aims at obtaining the critical oint of I on the whole sace from the local minimizer for I on Nehari manifold. Lemma Suose that u is a local minimizer for I on N + (or N ). Then I (u) = 0. Proof. If u 0, u is a local minimizer for I on N + (or N ), then u is a nontrivial solution of the otimization roblem minimize I subject to Ψ(u) = 0, where Ψ(u) is described in (2.5). Note that Ψ (u) 0, N + (or N ) is a local differential manifold. So by the theory of Lagrange multiliers, there exists µ R such that I (u) = µψ (u). Thus I (u), u = µ Ψ (u), u. Since u N + (or N ), I (u), u = 0 and Ψ (u), u = K u(1) 0. Hence, µ = 0. Thus the roof is comlete.

11 EJDE-2016/301 KIRCHHOFF TYPE PROBLEMS Proof of Theorem 1.1 For roving Theorem 1.1, we first show that any Palais-Smale sequence of I has a strongly convergent subsequence in E. Lemma 3.1. Each Palais-Smale sequence {u n } N + (or N ) for I on E has a strongly convergent subsequence. Proof. Assume that {u n } N + (or N ) such that I(u n ) c and I (u n ) 0 as n. As in Lemma 2.6, we know the Palais-Smale sequence {u n } N + (or N ) for I on E is bounded. And by Lemma 2.2, going if necessary to a subsequence, we have Note that (I (u n ) I (u), u n u) u n u in E, u n u in L r ( ), r [2, 2 ). = (I (u n ), u n u) (I (u), u n u) ( ) = a + b u n 2 dx (u n u) 2 dx + R ( 3 ) V (x) u n u 2 dx b u 2 dx u n 2 dx u (u n u) dx f(x)( u n q 2 u n u q 2 u)(u n u) dx R 3 g(x)( u n 2 u n u 2 u)(u n u) dx ( ) u n u 2 b u 2 dx u n 2 dx u (u n u) dx f(x)( u n q 2 u n u q 2 u)(u n u) dx R 3 g(x)( u n 2 u n u 2 u)(u n u) dx, then we can deduce that u n u 0 as n. Indeed, from the boundedness of {u n } in E and Lemma 2.2, {u n } is bounded in L r ( ), r [2, 6). By using twice Hölder inequality we obtain f(x)( u n q 2 u n u q 2 u)(u n u) dx R ( 3 ) f q 1/q ( dx u n q 2 u n u q 2 u /q u n u /q dx C f q ( u n q 1 + u q 1 ) u n u 0, as n, where C is a ositive constant. Similarly, we have g(x)( u n 2 u n u 2 u)(u n u) dx 0, as n. ) q/

12 12 X. CAO, J. XU, J. WANG EJDE-2016/301 From ( ) b u 2 dx u n 2 dx u (u n u) dx 0, as n, and (I (u n ) I (u), u n u) 0, as n, we have u n u 0 as n. This comletes the roof. Lemma 3.2. If f q g (2 q)/( 2) (0, σ), then the minimization roblem c 1 = inf N +I is solved at a oint u 1 N +, moreover this value is a critical oint of I. Proof. First we rove the minimizing sequence {u n } N + is a (P S) c1 sequence on E. Indeed, by Lemma 2.6 and Ekeland Variational Princile [35] on N + N 0, there exists a minimizing sequence {u n } N + N 0 such that inf I(u) I(u n) < inf I(u) + 1 u N + N 0 u N + N 0 n, (3.1) I(u n ) 1 n v u n I(v), v N + N 0. (3.2) From Proosition 2.5, we know for each u E\{0}, there is a unique t + such that t + u N +, then inf u N +I I(t + u). Now we rove that for each u N +, I(u) < 0. Indeed, for each u N +, K u(1) > 0. From (2.7), we have ( q) f(x) u q dx > ( 2) u 2 + ( 4)b u 4 2. Then for each u N +, I(u) = I(u) 1 I (u), u = 2 2 u b u 4 2 q f(x) u q dx q < 2 2 u b u ( ( 2) u 2 + ( 4)b u 4 q 2) = ( 2)(q 2) u 2 + 2q ( 4)(q 4) b u 4 2 < 0. 4q (3.3) From the above, we know that inf u N + I(u) < 0. Since I(0) = 0, we have inf u N + N 0 I(u) = inf u N + I(u) = c 1. Thus we may assume u n N +, I(u n ) c 1 < 0. By Lemma 2.8, since f q g (2 q)/( 2) (0, σ), we can find ɛ n > 0 and differentiable function ϕ + n = ϕ + n (w) > 0 such that ϕ + n (w)(u n w) N +, w B ɛn (0). By the continuity of ϕ + n (w) and ϕ + n (0) = 1, without loss of generality, we can assume ɛ n is sufficiently small such that 1 2 ϕ+ n (w) 3 2 for w < ɛ n. From ϕ + n (w)(u n w) N + and (3.2), we have which imlies I(ϕ + n (w)(u n w)) I(u n ) 1 n ϕ+ n (w)(u n w) u n, I (u n ), ϕ + n (w)(u n w) u n + o ( ϕ + n (w)(u n w) u n )

13 EJDE-2016/301 KIRCHHOFF TYPE PROBLEMS 13 Consequently, 1 n ϕ+ n (w)(u n w) u n. ϕ + n (w) I (u n ), w + (1 ϕ + n (w)) I (u n ), u n 1 n ( ϕ + n (w) 1 ) u n ϕ + n (w)w + o ( ϕ + n (w)(u n w) u n ). By the choice of ɛ n and 1/2 ϕ + n (w) 3/2, we infer that there exists C 3 > 0 such that I (u n ), w 1 n (ϕ+ n ) (0), w u n + C 3 n w + o ( (ϕ + n ) (0), w ( u n + w ) ). Next we rove for {u n } N +, inf n u n C 1 > 0, where C 1 is a constant. Indeed, if not, then I(u n ) would converge to zero, which contradict with I(u n ) c 1 < 0. Moreover, by Lemma 2.6 we know that I is coercive on N +, {u n } is bounded in E. Thus, there exists C 2 > 0 such that 0 < C 1 u n C 2. From Lemma 2.8, (ϕ + n ) (0), w C w. So and I (u n ), w C n w + C w + o( w ) n I (u n ) = I (u n ), w su C w E\{0} w n + o(1), I (u n ) 0, as n. (3.4) Thus, {u n } N + is (P S) c1 for I on E. From Lemma 3.1, there is a strongly convergent subsequence {u n }, we still denote by {u n }, u n u 1 in E. From the above we know that there exist C 1, C 2 > 0 such that 0 < C 1 u n C 2, then 0 < C 1 u 1 C 2. Thus u 1 0. Next we rove u 1 N +. Indeed, by (2.6), it follows that K u n (1) K From K u n (1) > 0, we have K u 1 (1) 0. By Proosition 2.4, we know K Thus we deduce u 1 N +, I(u 1 ) = lim I(u n) = inf I(u). n u N + u 1 (1). u 1 (1) > 0. We recall [16] that u 2 dx = u 2 dx, therefore I(u 1 ) = I( u 1 ) and u 1 N +, then without loss of generality we may assume that u 1 is ositive. Combining this with Lemma 2.10, we obtain the results. Lemma 3.3. If f q g (2 q)/( 2) (0, σ), then the minimization roblem c 2 = inf N I is solved at a oint u 2 N which is a critical oint for I. Proof. From Lemma 2.7, N is closed in E. By Lemma 2.6, we know I is coercive on N. So we use Ekeland Variational Princile [35] on N to obtain a minimizing sequence {u n } N such that inf I(u) I(u n ) < inf I(u) + 1 u N u N n, I(u n ) 1 n v u n I(v), v N.

14 14 X. CAO, J. XU, J. WANG EJDE-2016/301 In view of (2.15) and Lemma 2.6 we know the there exist C 1, C 2 > 0 such that 0 < C 1 u n C 2. Hence by Lemma 2.9, in the same way as Lemma 3.2, there exists a minimizing sequence {u n } N is the (P S) c2 sequence on E. From Lemmas 3.1, we know there is a strongly convergent subsequence {u n }, we still denote by {u n }, u n u 2 in E. By Lemma 2.7 the set N is closed, we know u 2 N. Thus, I(u 2 ) = lim n I(u n ) = inf u N I(u). Since I(u 2 ) = I( u 2 ) and u 2 N, then without loss of generality we may assume that u 2 is ositive. Combining with Lemma 2.10, we rove the claim. Now we are in a osition to give the roof of the main results. Proof of theorem 1.1. From Lemmas 3.2 and 3.3, we know if f q g (2 q)/( 2) (0, σ), then roblem (1.1) has at least two ositive solutions u 1 and u 2. From the roof of Lemma 3.2 we know that if f q g (2 q)/( 2) (0, σ), then the ositive solution of (1.1) u 1 belongs to N + and I(u 1 ) < 0. If 0 < f q g (2 q)/( 2) < σ := σ < σ, where σ is described in Proosition 2.4, then by (2.15) we can infer that q 2 I(u) = I(u) 1 4 I (u), u = 1 4 u 2 ( 1 q 1 4) 1 4 u 2 ( 1 q 1 4 = u q( 1 f(x) u q dx + ( 1 4 ) 1 g(x) u dx ) f q S q/2 u q 4 u 2 q ( 1 q 1 ) 4 ) f q S q/2 ( (2 q)s 2 ) q ( q) g ( (2 q)s 2 ) q ( q) g ( ( ( (2 q)s 2 ) 2 q 2 ( q) g ( (2 q)s 2 ) 2 q 2 ( q) g ( 1 q 1 4 q 4q ) ) f q S q/2 f q S q/2 ) > 0. In fact, if f q g (2 q)/( 2) (0, σ ), for any u N, I(u) > 0, where σ = q/( 2)σ. From Lemma 3.3 the ositive solution of roblem (1.1) u 2 N, (0, σ ), I(u 2 ) > 0. From the above, we know that if f q g (2 q)/( 2) (0, σ ), then I(u 1 ) = inf u N I(u), and u 1 is a ositive ground state solution of (1.1). This comletes the roof. then for f q g (2 q)/( 2) Acknowledgments. This work is suorted by the Innovation Project for college ostgraduates in Jiangsu Province(KYLX ), the Natural Science Foundation of China (Nos: , , , ), the Natural Science Foundation of Jiangsu Province (Nos: BK , BK , BK ) and the Natural Science Foundation of Outstanding Young Scholars of Jiangsu Province (Nos: BK ). References [1] A. Ambrosetti, H. Brezis, G. Cerami; Combined effects of concave and convex nonlinearities in some ellitic roblems, J. Funct. Anal., 122 (1994),

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