Xianhua Tang Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: −(a + b ∫ R3 |∇u|2 dx)∆u + V(x)u = (Iα ∗ F(u))f(u), x ∈ R3, u ∈ H1(R3), where a, b > 0, Iα : R3 → R, with α ∈ (0, 3), is the Riesz potential, V ∈ C(R3, [0,∞)), f ∈ C(R,R) and F(t)=∫ 0 f(s)ds. By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f . Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case where f(u) = |u|q−2u, with q ∈ (1 + α/3, 2].

If we let α → 0 in (1.1), then it becomes formally the following Kirchhoff-type problem with local nonlinearity g = Ff : (1.4) which is related to the stationary analogue of the Kirchhoff equation Equation (1.5) is proposed by Kirchhoff [15] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. For more details on the physical aspects, we refer the readers to [2,3,7,8,26]. After Lions [21] proposed an abstract functional analysis framework to (1.4), it has received more and more attention from the mathematical community; there have been many works about the existence of nontrivial solutions to (1.4) and its fractional version by using variational methods, for example, see [4, 6, 9, 12, 13, 16-18, 22, 30, 31, 34, 35, 37, 40-42] and the references therein. A typical way to deal with (1.4) is to use the mountain-pass theorem. For this purpose, one usually assumes that g(t) is superlinear at t = 0 and super-cubic at t = ∞. In this case, if g further satisfies the monotonicity condition (G1) g(t)/|t| 3 is increasing for t ∈ ℝ \ {0}, via the Nehari manifold approach, He and Zou [13] obtained the first existence result on ground state solutions of (1.4). For the case where g(t) is not super-cubic at t = ∞, Li and Ye [17] proved that (1.4), with special forms V = 1 and g(u) = |u| p−2 u for 3 < p < 6, has a ground state positive solution by using a minimizing argument on a new manifold that is defined by a condition which is a combination of the Nehari equation and the Pohozaev equality. This idea comes from Ruiz [33], in which the nonlinear Schrödinger-Poisson system was studied. Later, by introducing another suitable manifold differing from [17], Guo [12] and Tang and Chen [37] improved the above result to (1.4) t is increasing on (0, ∞), and (G3) g ∈ C(ℝ, ℝ) and g(t)t+6G(t) |t|t is nondecreasing on (−∞, 0) ∪ (0, ∞), where G(t) = ∫ t 0 g(s) ds. respectively, and some standard growth assumptions.
Compared with (1.2) and (1.4), it is more difficult to deal with (1.1) which involves two nonlocal terms. In [23], Lü investigated the following special form of (1.1): where q ∈ (2, 3 + α), μ > 0 is a parameter and g(x) is a nonnegative steep potential well function. By using the Nehari manifold and the concentration compactness principle, Lü proved the existence of ground state solutions for (1.6) if the parameter μ is large enough. It is worth pointing out that the same result is not available in the case where q ∈ (1 + α/3, 2], even when g(x) = 0, since both the mountain pass theorem and the Nehari manifold argument do not work. In fact, in this case, it is more difficult to get a bounded (PS) sequence and to prove that the (PS) sequence converges weakly to a critical point of the corresponding functional in To the best of our knowledge, there seem to be no results dealt with this case in the literature. As for the related study of problem (1.1) involving the critical exponents, we refer to [25,32] for more details. Motivated by the above-mentioned papers, we shall deal with the existence of ground state solutions for (1.1) under (V1) and (F1). It is standard to check, according to (1.3), that under (V1) and (F1), the energy functional defined in is continuously differentiable and its critical points correspond to the weak solutions of (1.1). We say a weak solution to (1.1) is a ground state solution if it minimizes the functional Φ among all nontrivial weak solutions.
In addition to (F1), we also need the following assumptions on f : is weaker than the following assumption, which is easier to verify: It is easy to see that there are many functions which satisfy (F1), (F2) and (F4). In addition, there are some functions which satisfy (F3), but not (F4), for example, To overcome the lack of compactness of Sobolev embeddings in unbounded domains, different from [23] in which a steep potential well was considered, we assume that V satisfies (V1) and the decay condition: (V2) V ∈ C 1 (ℝ 3 , ℝ) and either of the following cases holds: Next, we further provide a minimax characterization of the ground state energy. To this end, we introduce a new monotonicity condition on V as follows: (V3) V ∈ C 1 (ℝ 3 , ℝ) and there exists θ ∈ [0, 1) such that t → 4V(tx) + ∇V(tx) ⋅ (tx) + θa 2t 2 |x| 2 is nonincreasing on (0, +∞) for every x ∈ ℝ 3 \ {0}. Similar to [12], we define the Pohozaev functional related with (1.1): It is well known that any solution u of (1.1) satisfies P(u) = 0. Motivated by [17,37], we define the Nehari- Then every nontrivial solution of (1.1) is contained in M. Our second main result is as follows.
where and in the sequel u t (x) := u(x/t).
Applying Theorem 1.3 to the "limiting problem" of (1.1): and where P ∞ (u) = 0 is the Pohozave type identity related with (1.10). Then we have the following corollary.
To prove Theorem 1.2, we will use Jeanjean's monotonicity trick [14], that is, an approximation procedure to obtain a bounded (PS)-sequence for Φ, instead of starting directly from an arbitrary (PS)-sequence. More precisely, firstly, for λ ∈ [1/2, 1], we consider a family of functionals Φ λ : These functionals have a mountain pass geometry, and we denote the corresponding mountain pass levels by c λ . Moreover, Φ λ has a bounded (PS)-sequence {u n (λ)} ⊂ H 1 (ℝ 3 ) at level c λ for almost every λ ∈ [1/2, 1]. Secondly, we use the global compactness lemma to show that the bounded sequence {u n (λ)} converges weakly to a nontrivial critical point of Φ λ . To do this, we have to establish the following strict inequality: where Φ ∞ λ is the associated limited functional defined by However, it seems to be impossible to obtain the w ∞ λ mentioned above only under (F1)-(F3). So the usual arguments cannot be applied here to prove (1.14). To overcome this difficulty, we follow a strategy introduced in [38], that is, we first show that there existsū ∞ such that 16) and then, by means of the translation invariance forū ∞ and the crucial inequality and P ∞ λ (u) = 0 is the corresponding Pohozave type identity. In particular, any information on sign ofū ∞ is not required in our arguments. Finally, we choose two sequences {λ n } ∈ (λ , 1] and {u λ n } ⊂ H 1 (ℝ 3 ) \ {0} such that λ n → 1 and Φ λ n (u λ n ) = 0, and by using (1.17) and the global compactness lemma, we get a nontrivial critical pointū of Φ.
We would like to mention that in the proof of Theorem 1.2, a crucial step is to prove (1.16), which is a corollary of Theorem 1.3. Inspired by [5,36,37], we shall prove Theorem 1.3 by following this scheme: Step 1: we verify M ̸ = 0 and establish the minimax characterization of m := inf M Φ > 0, Step 2: we prove that m is achieved, Step 3: we show that the minimizer of Φ on M is a critical point.
Although we mainly follow the procedure of [36], we have to face many new difficulties due to the mutual competing effect between (∫ ℝ 3 |∇u| 2 dx)∆u and (I α * F(u))f(u). These difficulties enforce the implementation of new ideas and techniques. More precisely, in step 1, we first establish a key inequality, namely, Throughout the paper we make use of the following notations: • H 1 (ℝ 3 ) denotes the usual Sobolev space equipped with the inner product and norm The rest of the paper is organized as follows. In Section 2, we study the existence of ground state solutions for (1.1) by using the Nehari-Pohozaev manifold M, and give the proof of Theorem 1.3. In Section 3, based on Jeanjean's monotonicity trick, we consider the existence of ground state solutions for (1.1), and complete the proof of Theorem 1.2. In Section 4, we study the non-existence of solutions for problem (1.12) and present the proof of Theorem 1.5.

Proof of Theorem 1.3
In this section, we give the proof of Theorem 1.3. To this end, we give some useful lemmas. Since V(x) ≡ V ∞ satisfies (V1)-(V3), all conclusions on Φ are also true for Φ ∞ in this paper. For (1.4), we always assume that V ∞ > 0. First, by a simple calculation, we can verify the following lemma.  Proof. It is evident that (2.2) holds for all s > 0 and t = 0. For t ̸ = 0, it follows from (F3) that which implies that g(s, t) ≥ g(1, t) = 0 for all s ≥ 0 and t ∈ (−∞, 0) ∪ (0, +∞).

Note that
From Lemma 2.5, we have the following two corollaries.
There are two possible cases.
Proof of Theorem 1.3. In view of Lemmas 2.9, 2.14 and 2.15, there existsū ∈ M such that This shows thatū is a ground state solution of (1.1).

Proof of Theorem 1.2
In this section, we give the proof of Theorem 1.2. Without loss of generality, we consider that Then, for almost every λ ∈ K, there is a bounded (PS) c λ sequence for I λ , that is, there exists a sequence such that (i) {u n (λ)} is bounded in X, (ii) I λ (u n (λ)) → c λ , (iii) I λ (u n (λ)) → 0 in X * , where X * is the dual of X. Moreover, c λ is nonincreasing and left continuous on λ ∈ [1/2, 1].