Canlin Gan’s research while affiliated with Xidian University and other places

The Klein-Gordon-Maxwell system has received great attention in the community of mathematical physics. Under a special superlinear condition on the nonlinear term, the existence of solution for the critical Klein-Gordon-Maxwell system with a steep potential well has been solved. In this paper, under two general superlinear conditions, we obtain the existence of ground state solution for the critical Klein-Gordon-Maxwell system with a steep potential well. The general superlinear conditions bring challenge in proving the boundedness of Cerami sequence, which is a key step in the proof of the existence. To solve this, we construct a Pohožaev identity and adopt some analytical techniques. Our results extend the previous results in the literature.

The following kind of Klein–Gordon–Maxwell system is investigated −Δu+V(x)u−(2ω+ϕ)ϕu=K(x)f(u),inR3,Δϕ=(ω+ϕ)u2,inR3,\begin{equation*} \hspace*{4pc}{\left\lbrace \begin{aligned} &{-\Delta u+ V(x) u-(2\omega +\phi ) \phi u=K(x)f(u)}, & & {\quad \text{ in } \mathbb {R}^{3}}, \\ &{\Delta \phi =(\omega +\phi ) u^{2}}, & & {\quad \text{ in } \mathbb {R}^{3}}, \end{aligned}\right.} \end{equation*}where ω>0$\omega >0$ is a parameter, and V is vanishing potential. By using some suitable conditions on K and f, we obtain a Palais–Smale sequence by using Pohožaev equality and prove the ground‐state solution for this system by employing variational methods. Our result improves the related one in the literature.

This paper deals with the following system \begin{equation*} \left\{\begin{aligned} &{-\Delta u+ (\lambda A(x)+1)u-(2\omega+\phi) \phi u=\mu f(u)+u^{5}}, & & {\quad x \in \mathbb{R}^{3}}, \\ &{\Delta \phi=(\omega+\phi) u^{2}}, & & {\quad x \in \mathbb{R}^{3}}, \end{aligned}\right. \end{equation*} where $\lambda, \mu>0$ are positive parameters. Under some suitable conditions on A and f, we show the boundedness of Cerami sequence for the above system by adopting Poho\v{z}aev identity and then prove the existence of ground state solution for the above system on Nehari manifold by using Br\’{e}zis-Nirenberg technique, which improve the existing result in the literature.

In this paper, we study the Kirchhoff-type equation: where , , , and . and are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution to the above equation. Moreover, we obtain that the sign-changing solution has exactly two nodal domains. Our results can be seen as an improvement of the previous literature. 1. Introduction Consider the Kirchhoff-type equation where , , , and are continuous functions. Moreover, and are vanishing at infinity. Similar to [1], we say , if and satisfy the following conditions: (VQ1) , for any and (VQ2) If is a sequence of Borel sets such that their Lebesgue measures for all and some , then And one of the below conditions holds: (VQ3) or (VQ4) there exists such that Problem (1) is related to the following Kirchhoff equation Problem (4) is nonlocal due to the existence of which causes some mathematical analysis difficulties and makes problem (4) more interesting. The nonlocal operator comes from the Dirichlet problem where , , or is a bounded domain. Equation (5) is related to the following stationary analogue of the Kirchhoff-type equation: which was introduced by Kirchhoff [2] as generalization of the famous D’Alembert wave equation for free vibration of elastic strings. After Lion [3] investigated problem (5) involving an abstract framework, Kirchhoff-type equations have been extensively researched by many scholars. Hence, numerous interesting works to (5) or similar problems are obtained in the last decades. Please see [4–7] and the references therein. In particular, many scholars dedicated to searching sign-changing solutions to (4) and similar problems. Indeed, they obtained a lot of interesting results. For example, Li et al. [8] investigated the sign-changing solution to the following problem by using the constraint variational method They supposed that satisfies the following conditions: (f1) as uniformly in (f2) and for some , where and (f3) as uniformly in (f4) is an increasing function of . In [9], by using a similar main method and theorem, Wang, Zhang, and Cheng proved that if and satisfy (f1), (f3)-(f4) and (V1) such that for any (V2) (V3) there exist and is a nonincreasing function such that (f5) , then (4) has a sign-changing solution with two nodal domains. It is worth pointing out that they considered the associated “limiting problem” with the corresponding energy functional Moreover, they obtained , which is crucial to show that . Where For more interesting results, see [10–14]. When , , (4) reduces to Recently, (13) and similar problems have also been received far-reaching research. For example, in [15], when , , Chen considered (13) on a bounded domain and obtained the sign-changing solution to (13) under some sufficient hypothesis. Precisely, they assumed that satisfies the following conditions: (A1) , and as uniformly in (A2) , and as uniformly in (A3) is strictly increasing on for every (A4) as uniformly in (A5) , where Besides, letting be the first eigenvalue of . Then by (A5) and the definition of , they concluded . By using variational methods, Liu and Wang [16] obtained the least energy nodal solution to (13) in , when and , where is a parameter. They discussed two situations, the most important of which is that a nodal solution to (13) can be found for any under certain conditions, thanks to assumptions. (S1) and , such that where and are bounded smooth domains. (S2) where and , , . It is worth noting that Liu and Wang [16] used the first eigenfunction of , , when they showed that the constraint set is nonempty. For researching on problem (4) under conditions similar to [16], see [13]. For more relevant results, see [17–21] and the references therein. In fact, not only the sign-changing solutions of Kirchhoff-type equation has received widespread attention but also other kinds of solutions. Particularly, many scholars have investigated positive solutions, ground state solutions, multiple solutions, etc., for the Kirchhoff equation. Please see the literature [22–25]. Moreover, in recent years, there are also a lot of works on the following Kirchhoff-Schrdinger-Poisson systems Readers can refer to [26–32]. These results are also helpful for us to study (1), since (1) and (16) have a great correlation. Motivated by above-mentioned results, we will establish the existence of sign-changing solution to (1). Throughout this paper, we will make the following hypothesis about : (F1) if (VQ3) holds and for some if (VQ4) holds (F2) (F3) , where (F4) is nondecreasing for . Obviously, (F2)-(F3) are weaker than (f2)-(f3). In [9], the authors assume that satisfies (F1)-(F4), but they have more other assumptions on than this paper. This also makes the problem challenging and more interesting. Moreover, our hypothesis does not involve eigenvalue problems. Hence, our results can be regarded as the improvement and complementary of [15, 16]. In the paper, we use a direct method to obtain the sign-changing solution. In this paper, we consider our problem on the following space: with the norm Since , the embedding is compact for some (see Lemma 1), where endowed with the norm It is clear that is a Banach space. Let be the usual norm in . Hence, the embedding is continuous. Set Similar to [1], we can prove the following lemmas. Lemma 1. Let. If (VQ3) occurs, then the embeddingis continuous and compact. If (VQ4) satisfies, then the embeddingis continuous and compact. Lemma 2. Assume that. Ifandin, then Our main result is as follows. Theorem 3. Ifand (F1)-(F4) hold, then problem (4) has at least one least-energy sign-changing solution, which has precisely two nodal domains. Throughout this paper, , , denote positive constants possibly different in different places. The rest of this paper is organized as follows: in Section 2, some frameworks are demonstrated. In Section 3, the proof of the main result is given. 2. Preliminaries It is no doubt that the weak solution for (1) corresponds to the critical point of the energy functional Meanwhile, and we define And if is a solution to (1) with , then is called the sign-changing solution of (1), where Through a straightforward calculation, we obtain The only thing needed to obtain a sigh-changing solution to (1) is to find a minimizer of the energy of over the following constraint: For proving our result, we will adopt a regular process. First, we need to obtain the existence of a unique pair such that and is the unique maximum point of . That is to prove the following lemma. Lemma 4. Let. Suppose that (F1)-(F4) hold, if with , then there exists a unique pair of positive numbers such that and Proof. Set with . Letting First, we will show that there exists such that Case 1. If (VQ2) holds, from (F1)-(F2), for each , there exists a such that Then by Sobolev imbedding theorem, one has Case 2. If (VQ4) holds, refer to the argument in [1], there exists , such that Then by (F1)-(F2), we deduce that According to (34), (VQ1) and Hlder inequality, we obtain From the above discussion and , we can conclude that there is enough small such that Due to , then such that Besides, it follows from (F3) that for all , there exists such that Hence, as satisfies , we get Assume . We have And once again, if , we have or And Obviously, we can take large enough so that By (33), (34), and (41)–(45), we can deduce that (30) and (31) hold. Then from the point of view of Maranda’s theorem [2], there exists a pair of positive numbers such that Next, we will prove the uniqueness of . Without loss of generality, the only thing we need to do is to show that which makes is unique. Assume that there exists another pair such that . Let , by the definition of , we have Combining (47) and (48), we have Thanks to (F4) and , the right side of the above equality is positive, which is a contradiction since the left is negative. So must be equal to , that is, is unique. Similarly, we can show the uniqueness of . In short, the pair is unique. Next, we prove that . Letting . It is obvious that attains the unique critical value at in . From (F3), we obtain which implies that a maximum point of cannot be established on the boundary of . If we may assume that is a maximum of , for small enough, we can easily obtain that Then for small enough, is an increasing function with respect to . Hence, according to our discussion, the pair is not a maximum point of in . Remark 5. For proving the uniqueness of the pair , we can also discuss it in two situations. Case 1. . We need to show that . The process of proving in Case 1 shall be similar to the proof in Lemma 4. Case 2. . By Lemma 4, there exists a pair of positive numbers such that when . Assume that there exists another pair such that . Set and . Then, we have which implies and . Since and by Case 1, we consider the minimization problem Lemma 6. Assume that (F1)-(F4) hold and, then is achieved. Proof. Step 1. . For each , we have . From (33) and (36), we derive that or Choosing . It is easy to verify that there exists such that . According to (F4), we have Then, i.e., . Step 2. is achieved. Let such that . Then, is bounded in . Consequently, there exist such that in . Due to , then . That is Similar to (57), we can find a such that for any . Moveover, according to (F1)-(F2), for every , there is such that Therefore, from (58), we conclude Based on the boundedness of , there is such that Selecting , by (61), we have From (62) and the compactness of embedding for , then we have i.e., . According to the weak semicontinuity of the norm, we have By (58), (64), and Lemma 2, we have i.e., Then, there exists such that . Suppose that , then From (65) and (66), we have Similar to the discussion in Lemma 4, by (67), we obtain . Letting . Then by , we obtain (68) implies that . Then, and . The proof is completed. Lemma 7. Assume (F1)-(F4) hold, thenis a critical point of. Proof. Due to , then . By Lemma 4, for any and , we have Assume , then there exists and such that for every and . Let and , , where . Then, from Lemma 4, we have For , Lemma 6 in [2] shows that there is a deformation such that (i) if (ii)(iii) for all .We claim that By Lemma 4 and (iii), we have , i.e., . Moveover, from the definition of , it follows that which implies that for all . So . Then by (ii), we can obtain that (71) holds. Next, we will show that . Letting , By direct calculation, we get From (F4), we have Then, Let then . Therefore, is the unique isolated zero point of . Then by degree theorem, we have that . From (70) and (i), we have . Hence, . Then, there is such that . Hence, , which is a contradiction with (71). 3. Proof of Theorem 3 Proof of Theorem 3. From Lemmas 6 and 7, there is a such that and . Next, we will prove that has exactly two nodal domains. By a contradiction, let , and , where Letting . Then , , that is, . By Lemmas 4 and 6, there exists a unique pair such that . Then, Therefore, we have On the other hand, we have which is impossible. Hence, has two exactly two nodal domains. The proof is completed. Data Availability No data were used to support this study. Conflicts of Interest The authors declare that they have no competing interests. Authors’ Contributions The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript. Acknowledgments This work is supported by the National Natural Science Foundation of China (No. 11961014 and No. 61563013) and Guangxi Natural Science Foundation (2016GXNSFAA380082 and 2018GXNSFAA281021).

The existence of nontrivial solutions for the following kind of Klein–Gordon–Maxwell system $\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u-(2\omega +\phi )\phi u=f(x,u),&{}\quad x\in {{\mathbb {R}}}^{3},\\ \Delta \phi =(\omega +\phi )u^{2},&{}\quad x\in {{\mathbb {R}}}^{3}, \end{array}\right. \end{aligned}$is investigated, where $\omega >0$ is a constant, $V\in C({{\mathbb {R}}}^{3},{{\mathbb {R}}})$ is either periodic or coercive and is allowed to be sign-changing, $f\in C({{\mathbb {R}}}^{3}\times {{\mathbb {R}}},{{\mathbb {R}}})$ and f is subcritical and local super-linear. Using local super-quadratic conditions and other suitable assumptions on the nonlinearity f(x, u) and the potential V(x), the existence of nontrivial solutions for the above system is established. The obtained results in this paper improve the related ones in the literature.

In this paper, we intend to study the following Klein‐Gordon‐Maxwell system: −Δu+(λA(x)+1)u−(2ω+φ)φu=f(u),x∈ℝ3,Δφ=(ω+φ)u2,x∈ℝ3. By adopting some new analytical skills and employing variational methods, the existence of a ground state solution for the above system is established under suitable conditions on A and f . The range of ω is more wider than Liu et al., Theorem 1.1

This paper is concerned with the following system: {−Δu+λA(x)u−K(x)(2ω+ϕ)ϕu=f(x,u)+h(x),x∈R3,Δϕ=K(x)(ω+ϕ)u2,x∈R3, where λ≥1 is a parameter, ω>0 is a constant and the potential A is sign-changing. Under the classic Ambrosetti–Rabinowitz condition and other suitable conditions, nontrivial solutions are obtained via the linking theorem and Ekeland’s variational principle. Especially speaking, we use a super-quadratic condition to replace the 4-superlinear condition which is usually used to show the existence of nontrivial solutions in many references. Our results improve the previous results in the literature.