Multiple solutions for Kirchhoff type problems involving super-linear and sub-linear terms

In this paper, we consider the multiplicity of solutions for a class of Kirchhoff type problems with concave and convex nonlinearities on an unbounded domain. With the aid of Ekeland’s variational principle, Jeanjean’s monotone method and the Pohožaev identity we prove that the Kirchhoff problem has at least two solutions.


Introduction
This paper concerns the multiplicity of solutions for the following Kirchhoff type problem where a, b are positive constants, 1 < q < 2, g(x) is a continuous function and f is a superlinear, subcritical nonlinearity.Kirchhoff type problems were proposed by Kirchhoff in 1883 [14] 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 the length of the string produced by transverse vibrations.It is related to the stationary analogue of the equation where u denotes the displacement, h(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).Such problems are often viewed as nonlocal because the presence of the integral term Ω |∇u| 2 dx which implies that the problem (1.2) is no longer a pointwise identity.This phenomenon causes some mathematical difficulties making the study of such problems particularly interesting.
Besides, a similar nonlocal problem also appears in other fields such as physical and biological systems, where u describes a process that depends on its average, for example, the population density.
The case of Kirchhoff problems where the nonlinear term is super-triple or super-linear has been investigated in the last decades by many authors, for example [4-10, 13, 17-20] and references therein.Here, we are interested in the case of Kirchhoff problems where the nonlinearity includes super-linear and sub-linear terms.Recently, Chen and Li [3] considered the following nonhomogeneous Kirchhoff equation where a, b > 0, N ≤ 3.Under the conditions g(x) ∈ L 2 (R N ) and where C 1 is a positive constant and "meas" denotes the abbreviation of Lebesgue measure in R N ; Jiang, Wang and Zhou [12] studied the following nonhomogeneous Schrödinger-Maxwell system where λ > 0, p ∈ (2, 6) and 0 ≤ g(x) = g(|x|) ∈ L 2 (R 3 ).By using a cut-off functional to obtain a bounded Palais-Smale sequence ((PS) sequence in short), they proved that there is a constant C p > 0 such that (1.4) has at least two solutions for p ∈ (2, 6) provided that g L 2 ≤ C p ; however, for p ∈ (2, 3) they needed to assume in addition that λ > 0 is small.Li, Li and Shi [15] considered the following Kirchhoff type problem where N ≥ 3, a, b > 0 and the parameter λ ≥ 0. Under the conditions ( f 1 ) and they proved that there exists λ 0 > 0 such that for any λ ∈ [0, λ 0 ), (1.5) has at least one positive solution.Moreover, they pointed out that it is not clear whether (1.5) has a solution for large Motivated by these papers [3,8,12,15,17], we consider the Kirchhoff problem (1.1) with super-linear and sub-linear terms on the whole space R 3 .By the fact that the nonlocal term ( R 3 |∇u| 2 dx) 2 is homogeneous of degree 4 and that the nonlinearity is a combination of super-linearity and sub-linearity, we are unable to use the method in [3] to obtain a bounded (PS) sequence.Here, we overcome the difficulties with the aid of Jeanjean's monotone method and the Pohožaev identity.Theorem 1.1.Assume that in the problem (1.1), f (u) = |u| p−2 u with 2 < p < 6 and g(x) is a nonnegative function with the following property: (g 1 ) 0 ≤ g(x) = g(|x|) = 0 and g(x) ∈ C 1 (R 3 ) ∩ L q * (R 3 ), where q * = 2 2−q ; (g 2 ) ∇g(x), x ∈ L q * (R 3 ).
There exists σ > 0 such that if |g| q * ∈ (0, σ), the problem (1.1) has two positive solutions, one of which has a positive energy and the other a negative energy.
Remark 1.4.In the previous papers, because of the nonlocal term ( R 3 |∇u| 2 dx) 2 with 4degrees, the Kirchhoff problem (1.1) is usually considered under the condition ( f 3 ), which implies that f (u) is super-triple.In other way, the nonlinear condition ( f 3 ) demands N ≤ 3, thus the corresponding Kirchhoff type problem is usually studied in R N with N ≤ 3.In the spirit of [3,8,12,15,17], we consider the Kirchhoff problem (1.1) under the condition ( f 4 ), which implies the nonlinearity f (u) is a super-linear term.This nonlinear condition ( f 4 ) can allow the dimension N ≥ 3.However, in this paper, for simplicity, we still consider the problem (1.1) in R 3 .
Remark 1.5.Consider the problem (1.1) with q = 1: where 0 ≤ g(x) = g(|x|) ∈ L 2 (R 3 ) and f satisfies the conditions ( f 1 )-( f 4 ).By a similar method we can prove that there is a constant C p > 0 such that if g L 2 ≤ C p , (1.6) has two solutions with different signs of the energies.
Remark 1.6.We can also consider the following Kirchhoff problem: where 1 ≤ q < 2. Suppose that V(x) satisfies the condition (V) and V(x) + ∇V(x), x satisfies suitable condition; g(x) is a continuous function and satisfies the conditions (g 1 )-(g 2 ) or (g 1 )-(g 3 ); f is a superlinear and subcritical nonlinearity, which satisfies the conditions ( f 1 )-( f 4 ).With a similar method we can obtain similar results for (1.7).
This paper is organized as follows: Section 2 is dedicated to the abstract framework and some preliminary results.Sections 3 and 4 are concerned with the proofs of Theorems 1.1 and 1.2, respectively.
Throughout this paper, C or C i is used in various places to denote distinct constants.L p (R N ) denotes the usual Lebesgue space endowed with the standard norm When it causes no confusion, we still denote by {u n } a subsequence of the original sequence {u n }.

Preliminary results
In this section, we will recall some preliminaries and establish the variational setting for our problem.Since g is radially symmetric, we consider the problem in the radial space H 1 r (R 3 ), whose compactness is very important to our proof.Let E = H 1 r (R 3 ) be the subspace of H 1 (R 3 ) consisting of the radial functions and equipped with the norm which is equivalent to the usual one for a > 0.
The energy functional corresponding to (1.1) is where F(u) = u 0 f (s)ds.It is well known that a weak solution of problem (1.1) is a critical point of the functional I.In the following, we are devoted to finding critical points of I.
First we give the following lemma.

Lemma 2.1 ([1, 22]).
The embedding E → L p (R 3 ) is continuous for p ∈ [2, 2 * ] and compact for p ∈ (2, 2 * ).Denote by S p the best Sobolev constant for the embedding E → L p (R 3 ), which is given by In particular, In what follows, we recall the following two lemmata, which play an important role in obtaining a bounded (PS) sequence of I.

Lemma 2.2 ([11]
).Let (X, • ) be a Banach space and J ⊂ R + be an interval.Consider the family of C 1 functionals on X of the form where B(u) ≥ 0 and either A(u) → ∞ or B(u) → ∞ as u → ∞.Assume that there are two points Then, for almost every λ ∈ J, there is a sequence {v n } ⊂ X such that Furthermore, the map λ → c λ is continuous from the left and non-increasing.
Lemma 2.3 (Pohožaev identity [2,12,16]).Let u ∈ H 1 (R 3 ) be a weak solution to the problem (1.1), then we have the following Pohožaev identity: x |u| q dx =: P(u). (2.2) 3 Proof of Theorem 1.1 In this section, we are devoted to the proof of Theorem 1.1, so we suppose that the assumptions of Theorem 1.1 hold throughout this section.First, we prove some useful preliminary results.
Lemma 3.2.If {u n } ⊂ E is a bounded (PS) sequence of I, then {u n } has a strongly convergent subsequence in E.
Proof.By Lemma 2.1, going if necessary to a subsequence, we have Note that From the boundedness of {u n } in E and Lemma 2.1, By twice using the Hölder inequality we obtain where C is a positive constant.Similarly, we have we have u n − u → 0 as n → ∞.This completes the proof.

Lemma 3.3.
There exists u 1 ∈ E such that where B α = {u ∈ E : u ≤ α} and α is given in Lemma 3.1.
Proof.We choose a function v ∈ E such that g(x)v(x) = 0, then for t > 0 small enough, we have This shows that c 1 := inf{I(u) : u ∈ B α } < 0. By Ekeland's variational principle [21], there exists {u n } ⊂ B α which is a bounded (PS) sequence of I.Then, by Lemma 3.2, there exists u 1 ∈ E such that u n → u 1 as n → ∞ in E. Hence I(u 1 ) = c 1 < 0 and I (u 1 ) = 0.
In order to apply Lemma 2.2 to get another solution, we introduce the following approximation problem: Then {I λ } λ∈J is a family of C 1 -functionals on E associated with the problem (3.2),where J = [ 1 2 , 1].Obviously, we have B(u) ≥ 0, ∀u ∈ E, and In the following lemma, we show that the family of functionals {I λ } satisfies the assumptions of Lemma 2.2.Lemma 3.4.If |g| q * < σ, then for any λ ∈ J the following conclusions hold.
Then, thanks to Lemmata 2.2, 3.2 and 3.4, there exists In view of Lemma 3.2, if the sequence {u n } ⊂ E given above is bounded, there exists u 2 = 0 such that I (u 2 ) = 0.In particular, u 2 is a non-trivial positive solution of the problem (1.1).
To complete the proof of Theorem 1.1, we just require that {u n } ⊂ E is bounded.Let From (3.3) and Lemma 2.3, we have Proof.We prove the lemma by the following two steps.
This together with the third equation of (3.6) implies that p = 6 + o n (1).So, if p = 6, (3.6) is impossible to hold.Thus, {|∇u n | 2 } is bounded.Now we are in a position to give the proof of Theorem 1.1.
Proof of Theorem 1.1.By Lemma 3.3, we find a solution u 1 of the equation (1.1) with negative energy.By Lemma 3.4, due to the Mountain Pass Theorem [21], we get a critical point u 2 of I corresponding to positive energy.Because u 1 and u 2 have different energies, it follows that u 1 = u 2 .Moreover, by strong maximum principle, u 1 and u 2 are positive.Thus, we obtain two positive solutions u 1 and u 2 , one of which corresponds to positive energy and another one negative energy.

Proof of Theorem 1.2
At first, we assume that the assumptions of Theorem 1.2 always hold in this section.Since f is a nonhomogeneous nonlinearity, the method of Lemma 3.5 is not available.However, by the condition (g 3 ), we can still obtain a bounded (PS) sequence.Before proving Theorem 1.2, we give some useful preliminary results.
In the same way as the previous section, we introduce the following approximation problem: Then {I λ } λ∈J is a family of C 1 -functionals on E corresponding to (4.3), where Similarly to Lemma 3.4, in the following lemma we want to show that {I λ } satisfies the assumptions of Lemma 2.2.Lemma 4.4.If |g| q * < σ, then for any λ ∈ J, the following conclusions hold.
To complete the proof of Theorem 1.2, it is sufficient to prove that {u n } is bounded in E.
Lemma 4.5.{u n } is bounded in E.
Proof of Theorem 1.2.By Lemma 4.3, we find a solution u 1 of the equation (1.1) with negative energy.By Lemma 4.4, due to the Mountain Pass Theorem [21], we get a critical point u 2 of I, whose energy is positive.Thus, u 1 and u 2 are two different solutions with their energies having different signs.If, in addition, f (u) is odd, the corresponding functional is even, then the solutions u 1 and u 2 are positive.

Lemma 4 . 2 .
If {u n } ⊂ E is a bounded (PS) sequence of I, then {u n } has a strongly convergent subsequence in E.