In this paper, we investigate the existence of a normalized solution for the following Kirchhoff system in the entire space ${\mathbb{R}}^N$ ( $N\ge 3$):

$$\begin{array}{r}\{\begin{array}{l}-\left(1+{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^{2}dx\right)\mathrm{\Delta }u={\lambda }_{1}u+{\mu }_1|u{|}^{p-2}u+\beta r_1|u{|}^{r_1-2}u|v{|}^{r_2},\\ -\left(1+{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }v{|}^{2}dx\right)\mathrm{\Delta }v={\lambda }_{2}v+{\mu }_2|v{|}^{q-2}v+\beta r_2|u{|}^{r_1}|v{|}^{r_2-2}v,\end{array}(\mathrm{P})\end{array}$$

under the constraints ${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=m_1\text{and}{\int }_{{\mathbb{R}}^N}|v{|}^{2}dx=m_2$, where $m_1,m_2>0$ are prescribed. The parameters ${\mu }_1,{\mu }_2,\beta >0$, $2\le p,q<2+\frac{8}{N}$, $r_1,r_2>1,$ and satisfy $r_1+r_2=2^{\ast }=\frac{2N}{N-2}$. The frequencies ${\lambda }_1,{\lambda }_2$ appear as Lagrange multipliers. With the help of the Pohožaev manifold and the minimization of the energy functional over a combination of the mass constraints and the closed balls, we obtain a positive ground state solution to (P). We mainly extend the results of Yang (Normalized ground state solutions for Kirchhoff-type systems) concerning the above problem from a single critical to a coupled critical nonlinearity.

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in ${\mathbb{R}}^N$ that involves a general double-phase variable exponent elliptic operator $\mathit{\varphi }$. Specifically, the operator $\mathit{\varphi }$ has behaviors like $|\tau {|}^{q(x)-2}\tau$ if $|\tau |$ is small and like $|\tau {|}^{p(x)-2}\tau$ if $|\tau |$ is large, where $1<p(x)<q(x)<N$. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature.

We investigate the normalized solutions for the following Schrödinger–Choquard equations constrained to $L^2$-sphere with general potential:

$$\begin{array}{r}\{\begin{array}{ll}-\mathrm{\Delta }v-V(x)v-\lambda v=(I_{\alpha }\ast F(v))f(v),& \mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}|v{|}^2\mathrm{d}x=a,& \mathrm{i}\mathrm{n}{\mathbb{R}}^N,\end{array}\end{array}$$

where $N\ge 3$, $a>0$ is a prescribed mass, $V\in C^1({\mathbb{R}}^N,\mathbb{R})$ is an external potential, $f\in C(\mathbb{R},\mathbb{R})$, $I_{\alpha }:{\mathbb{R}}^N\to \mathbb{R}$ denotes the Riesz potential with order $\alpha \in (0,N)$, and $\lambda \in \mathbb{R}$ is not prescribed in advance but appears as a Lagrange multiplier. By using critical point theory, we develop reliable arguments to construct the existence results of normalized solutions to this kind of Schrödinger–Choquard equations. The obtained results generalize and improve existing results in the literature.

In this paper, the existence of multiple solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory was investigated. The potential and the primitive of the nonlinearity in this kind of elliptic equations are both allowed to be sign-changing. Besides, we assumed that the nonlinearity satisfies the Berestycki–Lions type conditions. By employing Ekeland's variational principle, mountain pass theorem, Pohožaev identity, and various other techniques, two nontrivial solutions were obtained under some suitable conditions.