Improved results for Klein-Gordon-Maxwell systems with general nonlinearity

This paper is concerned with the following Klein-Gordon-Maxwell system \begin{document}$\left\{ \begin{align} &-\vartriangle u+\left[ m_{0}^{2}-{{(\omega +\phi )}^{2}} \right]u = f(u),\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ &\vartriangle \phi = (\omega +\phi ){{u}^{2}},\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ \end{align} \right.$ \end{document} where \begin{document}$0 and \begin{document}$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$\end{document} . By introducing some new tricks, we prove that the above system has 1) a ground state solution in the case when \begin{document}$0 and \begin{document}$f$\end{document} is superlinear at infinity; 2) a nontrivial solution in the zero mass case, i.e. \begin{document}$ω = m_0$\end{document} and \begin{document}$f$\end{document} is super-quadratic at infinity. These results improve the related ones in the literature.


(Communicated by Andrea Malchiodi)
Abstract. This paper is concerned with the following Klein-Gordon- in R 3 , where 0 < ω ≤ m 0 and f ∈ C(R, R). By introducing some new tricks, we prove that the above system has 1) a ground state solution in the case when 0 < ω < m 0 and f is superlinear at infinity; 2) a nontrivial solution in the zero mass case, i.e. ω = m 0 and f is super-quadratic at infinity. These results improve the related ones in the literature. 1. Introduction. In this paper, we study the following Klein-Gordon-Maxwell sys- where 0 < ω ≤ m 0 , u, φ : R 3 → R, and f : R → R satisfies the following basic assumptions: (F1) f ∈ C(R, R) and there exist constants C 0 > 0 and p ∈ (2, 6) such that |f (t)| ≤ C 0 1 + |t| p−1 , ∀ t ∈ R; (F2) f (t) = o(|t|) as t → 0. Such a system was first introduced by Benci and Fortunato [4] as a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. The presence of the nonlinear term f simulates the interaction between many particles or external nonlinear perturbations. For more details in the physical aspects, we refer the readers to [4,5]. Under assumptions (F1) and (F2), the weak solutions to (1.1) correspond to the critical points of the energy functional defined in H 1 (R 3 ) × D 1,2 (R 3 ) by
Their method consisted in minimizing the corresponding functional associated with (1.3) on the Nehari manifold. Following the ideas of [2], Wang [27] further extended the existence range of (m 0 , ω) for 2 < q < 4 as follows: Based on the Pohozaev identity and the "monotonicity trick", developed by Struwe [23] and Jeanjean [17], Azzollini et al. [3] proved that (1.3) has a nontrivial radial solution under one of the following conditions: (iii) 3 ≤ q < 4 and 0 < ω < m 0 ; (iv) 2 < q < 3 and 0 < ω < (q − 2)(4 − q) m 0 , which gave a little improvement for the existence results on (1.3) when 2 < q < 4. We point out that the approaches used in [2,3,27] are heavily dependent on the form f (t) = |t| q−2 t, which are no longer applicable for (1.1) with general nonlinearity f , even for the case when f (t) = |t| α−2 + |t| β−2 t with 2 < β < 4 and β < α < 6. Motivated by the aforementioned works, in the present paper, we will use some new tricks to generalize the above results to (1.1) under the following superlinear condition: (F3) there exist constants r 0 > 0 and µ > 2 such that Our first result is as follows.
2. If f (u) = |u| q−2 u with 2 < q < 6, then one has µ = q in (F3). In this case, Theorem 1.1 reduces to the best results (iii) and (iv) for (1.3) which was obtained in [3]. Therefore, Theorem 1.1 generalizes and improves the related ones in the literature.
Next, we are mainly interested to study the limit case ω = m 0 , when (1.1) becomes We notice that, in the first equation, besides the interaction term 2ωφ + φ 2 u, there is no linear term in u. In this sense the situation described by (1.4) is analogous to the zero mass case for nonlinear field equations (see [6]). In contrast to the case 0 < ω < m 0 , there are few papers dealing with the limit case ω = m 0 , we only find one paper [3]. More precisely, under the following assumption: (AQ) f ∈ C(R, R) and there exist constants 4 < q 0 ≤ q 1 < 6 < q 2 and C 1 , C 2 > 0 such that Azzollini et al. [3] proved that (1.4) has a nontrivial solution in D 1,2 (R 3 )×D 1,2 (R 3 ). Notice that the condition q 0 > 4 in (AQ) plays a crucial role. In the present paper, instead of (AQ), we will use the following weaker condition: (F4) f ∈ C(R, R) and there exist constants 3 < p 0 ≤ p 1 < 6 < p 2 and θ 1 , θ 2 > 0 such that This type of condition goes back to the work by Berestycki and Lions in [6], where the authors proved the following Schrödinger equation with zero mass has a nontrivial radial solution if g behaves like |t| p1−2 t for small t, and like |t| p2−2 t for large t with constants 0 We are now in a position to state the second result of this paper.
The paper is organized as follows. In Section 2, we give some preliminary lemmas. We prove Theorems 1.1 and 1.3 in Sections 3 and 4 respectively.
Throughout this paper, we let u t (x) := u(tx) for t > 0, denote the norm of L s (R 3 ) by u s = R 3 |u| s dx 1/s , and positive constants possibly different in different places, by C 1 , C 2 , · · · . 2336 SITONG CHEN AND XIANHUA TANG 2. Preliminary lemmas. Hereafter, H 1 (R 3 ) is the usual Sobolev space with the standard scalar product and norm As it has been done by the aforementioned authors, we apply the "reduction method" developed by Benci and Fortunato [5] in order to avoid the difficulty originated by the strongly indefiniteness of the functional S defined by (1.2). Now, we need the following technical results.
Multiplying (2.1) by φ u and integrating by parts, we obtain By the definition of S and using (2.3), the functional I(u) = S(u, φ u ) has the form In view of Lemmas 2.1 and 2.2, under (F1) and (F2), we have I ∈ C 1 (H 1 (R 3 ), R), and is a solution of (1.1) if and only if u is a critical point of I, and φ = φ u which is unique. For the sake of simplicity, in many cases we just say , is a weak solution of (1.1).
In view of [14, (3.25)], any critical point u of I satisfies the following Pohozaev equality: and To show M = ∅, we shall use the following general minimax principle [19, Proposition 2.8], which is a somewhat stronger variant of [28, Theorem 2.8].
Next, we will apply Lemma 2.3 to obtain a Cerami sequence {u n } of I with J(u n ) → 0. This idea goes back to Jeanjean [16].
Since I (u), u = 0 for u ∈ M, by (2.3), (2.5), (2.9) and Sobolev embedding inequality, one has which implies that there exists a constant 0 > 0 such that Thus, it follows from (2.42) and (2.43) that 3. Proof of Theorem 1.1. In this section, we give the proof of Theorem 1.1.