Ground state solutions for Kirchhoff-type equations with general nonlinearity in low dimension

where N = 1, 2,m : [0,∞) → (0,∞) is a continuous function, V :RN → R is differentiable, and f ∈ C(R,R). We obtain the existence of a ground state solution of Nehari–Pohozaev type and the least energy solution under some assumptions on V , m, and f . Especially, the existence of nonlocal termm(‖∇u‖2 L2(RN) ) and the lack of Hardy’s inequality and Sobolev’s inequality in low dimension make the problem more complicated. To overcome the above-mentioned difficulties, some new energy inequalities and subtle analyses are introduced.


Introduction
In this paper, we consider the following Kirchhoff-type equation: Problem (1.3) was proposed by Kirchhoff [20] when m(t) = a + bt and N = 1. We refer the readers to [2-4, 9, 10] for more mathematical and physical background on Kirchhoff-type problems.
Lions [25] proposed an abstract functional analysis framework to the Kirchhoff equation (1.4) after which extensive attention to (1.4) was aroused. In recent years, the following Kirchhoff-type problem has been studied intensively by many researchers, where V ∈ C(R N , R) and f ∈ C(R N × R, R), a, b > 0 are constants. By variational methods, a number of important results of the existence and multiplicity of solutions for problem (1.5) have been established with f satisfying various conditions. In the meantime, V is usually assumed as a constant, or periodic, or radial, or coercive, see for example [1, 5, 7, 8, 11, 14-16, 19, 21, 23, 24, 26-28, 30, 31, 34, 36-38] and the references therein. Recently, Ikoma [17] investigated the following Kirchhoff-type equations with power type nonlinearity: ⎧ ⎨ ⎩ -m( ∇u 2 L 2 (R N ) ) u + V (x)u = |u| q-1 u, x ∈ R N ; u ∈ H 1 (R N ). (1.6) Here, q ∈ (1, ∞) if N = 1, 2 and q ∈ (1, N+2 N-2 ) if N ≥ 3. Problem (1.6) in the general dimensions was considered, and the author proved that problem (1.6) has a ground state solution. Precisely, they assumed the following conditions on m and V when N = 1, 2: (M1) m ∈ C([0, ∞)) and there exists m 0 > 0 such that 0 < m 0 ≤ m(s) for all s ∈ [0, ∞); (M2) There exist q 0 > 0 and ε 0 > 0 such that M(s) -1 q 0 +1 m(s)s ≥ ε 0 s in (0, ∞); (M3) The function s -1 M(s) is nondecreasing in (0, ∞); (Ṽ3) Let q 0 be the constant that appeared in (M2). When 1 < q < 2q 0 + 1, there exist α, β > 0 such that Through refined topological analysis on energy functional of "limit problem", Ikoma compared the mountain pass level c v,λ with the one of the limit equation c ∞,λ . Specially, in Ikoma's analysis, the situation of N = 1 is different from the one when N ≥ 2. To get a ground state solution of (1.6), the author used the standard arguments which combined the monotonicity trick to get a bounded (PS) c v,λ sequence and the concentrationcompactness lemma to prove that the sequence has a convergent subsequence. It is worth noting that, in Ikoma's argument, he had to face the difficulties about obtaining the boundedness of the (PS) sequence and the strongly convergent subsequences when 1 < q < 2q 0 + 1.
Tang and Chen [32] dealt with the Schrödinger-Kirchhoff type equation when the nonlinearity f is more general. They proved that (1.7) possesses a ground state solution of Nehari-Pohozaev type and the least energy solution with the following assumptions about f , here N = 3: (F1) f ∈ C(R, R) and there exist constants C 0 > 0 and p 0 ∈ (2, 2 * ) such that |t| p-N+1 t is nondecreasing on t ∈ (-∞, 0) ∪ (0, ∞), where p > 2. Precisely, they proved that there exists the least energy solution following the standard approach as Ikoma's result. The difference is that they developed a new trick and compared the level c λ with the energy I ∞ λ of the minimizer u ∞ λ more directly. Here, c λ is the limited energy of a bounded (PS) sequence {u n (λ)} for almost every λ ∈ [1/2, 1). By using their original highlight inequalities to obtain the comparison, they got a minimizer u ∞ λ on the Nehari-Pohozaev manifold which is also used in Li [22] and Ruiz [29] for λ ∈ [1/2, 1]. Then, by using the global compactness lemma obtained in [22], they got a nontrivial critical point u λ which possesses energy c λ . Their approach is applicable to the problems of other types, such as Schrödinger-Poisson problems [32], Choquard equations [33], and so on.
Let us point out that some arguments and tricks used in [32] fail to adapt directly to oneand two-dimensional cases for Hardy's and Sobolev's inequalities and do not work at this point. In this sense, it is more complicated than a three-dimensional case. To the best of our knowledge, there are few results concerning (1.1) in one-and two-dimensional case. Based on [17,32], the main purpose of this paper is to extend and complement the corresponding existence results on (1.7) in a three-dimensional case and above to the lower dimensional situation. However, the general term m(t) is more difficult to deal with than the special form a + bt, and the appearance of m( ∇u 2 L 2 (R N ) ) makes problem (1.1) more intricate. In addition, it is noticed that there is no need to distinguish N = 1 and N = 2 in our approach. Now we state our hypotheses. Let N = 1, 2. For V , we suppose (V1), (V2) as above, and we use a mild hypothesis (V3) in place of (Ṽ3). Furthermore, (V4) makes some subtle inequalities hold, which helps us to prove the existence of ground states. For f , we suppose (F1)-(F4). For m, we assume (M1) above, (M2) instead of (M2), (M3), and (M4).
is nonincreasing on (0, +∞) for all x ∈ R N \{0}, where · denotes the inner product in R N , p > 2 holds here and after; A simple example of m satisfying all conditions (M1)-(M4) is the following: where a > 0, b > 0, p > 2. We can easily verify that m(t) fits all the above hypotheses.
To state our results, we define the norm in (1.9) and the energy functional Under assumptions (V1), (F1), and (F2), weak solutions to (1.1) correspond to critical points of E and E ∈ C 1 (R N , R). For any ε > 0, it follows from (F1) and (F2) that there exists C ε > 0 such that Let us define the Pohozaev functional for (1.1) by that is, P(u) = 0 if u is a critical point of E, and define the following constraint set: Our main results are as follows. The paper is organized as follows. In Sect. 2, we give some preliminaries and the proof of Theorem 1.1. Section 3 is devoted to proving Theorem 1.2. In this paper, C 1 , C 2 , . . . denote positive constants possibly different in different places.

Preliminaries
To obtain the ground state solution of (1.1), we establish the key energy inequality related with E and J. To this end, we first prove the following two inequalities.
From Lemma 2.3, we have the following corollary.
(2.23) (ii). Together a direct calculation with (V2), it follows from (2.3) that for all t ∈ R, x ∈ R N . There exists t 0 > 0 small enough such that This shows that I = inf u∈M E(u) > 0.
The following lemma has been proved in [12] and [32].

Lemma 2.9
Assume that (F1) and (F2) hold. If u n u in H 1 (R N ), then along a subsequence of {u n }, Similar to [31, Lemma 2.10], by using Lemma 2.9, we have the following lemma. To overcome the lack of the compactness of Sobolev embedding, we define its limit problem related to (1.1) by Since V (x) ≡ V ∞ , which is well covered by (V1), (V2), and (V4), then all the above conclusions on E are true for E ∞ . Through discussing the corresponding limit equation (2.32), we will get the proof of Theorem 1.1.

Lemma 2.12 Assume that (M1)-(M3) and (F1)-(F4) hold. Then I
Since J ∞ (u n ) = 0, then it follows from (2.27) that The same as the case of dimension three in [31], we get the following relation between I and I ∞ . Step This shows thatū is a ground state solution of Nehari-Pohozaev type for (1.1).

The least energy solutions for (1.1)
In this section, we give the proof of Theorem 1.2.

Proposition 3.1 ([18]
) Let X be a Banach space and ⊂ R + be an interval. We consider a family { λ } λ∈ of C 1 -functionals on X of the form where B(u) ≥ 0, ∀u ∈ X, and such that either A(u) → +∞ or B(u) → +∞, as u → ∞. We assume that there are two points v 1 , v 2 in X such that Then, for almost every λ ∈ , there is a bounded (PS) c λ sequence for λ , that is, there exists a sequence such that To apply Proposition 3.1, we introduce two families of functionals defined by and We set J λ (u) := E λ (u), u + P λ (u), then for λ ∈ [1/2, 1]. Correspondingly, we also let By Lemma 2.3, we have the following lemma.
The proof of Lemma 3.4 is standard, the reader can refer to [6,Lemma 4.4].
Analogous to the proof of Lemma 2.3 in [22], we can prove Lemma 3.6. We omit it here.