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We study a class of $p(x)$-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.

We prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_\Omega \frac{1}{p(x)}|\nabla \Delta u|^{p(x)}dx)\Delta^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) \ \ \mbox{on} \ \Omega$ and $ \ u=\Delta u=\Delta^2 u=0 \ \ \mbox{on} \ \partial\Omega,$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $\Delta_{p(x)}^3u = \operatorname{div}\Big(\Delta(|\nabla \Delta u|^{p(x)-2}\nabla \Delta u)\Big)$ is the $p(x)$-triharmonic operator, $p,q,r \in C(\overline\Omega)$, $1< p(x) < \frac N3$ for all $x\in \overline\Omega$, $M(s) = a - bs^\gamma$, $a,b,\gamma>0$, $\lambda>0$, $g: \Omega \times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function while $f,h : \Omega \times \mathbb{R} \to \mathbb{R}$ are sign-changing continuous functions in $\Omega$. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order $p(x)$-Kirchhoff type problems with sign changing Kirchhoff functions.

We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-\Delta)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in \Omega, \\ \\ u= 0, \quad x\in \partial\Omega, \end{array}\right.$$ where $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overline{\Omega}\times \overline{\Omega} \rightarrow (1, \infty)$, $(-\Delta)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,\infty) \to [0, \infty),$ and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo-Benci-Fortunato (Nonlinear Anal. 7:9 (1983), 981-1012), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti-Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

In this paper, we study the following $p(x)$-curl systems: \begin{eqnarray*} \begin{cases} \nabla\times(|\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u})+a(x)|\mathbf{u}|^{p(x)-2}\mathbf{u}=\lambda f(x,\mathbf{u})+\mu g(x,\mathbf{u}),\quad\nabla\cdot \mathbf{u}=0,\; \mbox{ in } \Omega, \\ |\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u}\times \mathbf{n}=0,\quad \mathbf{u}\cdot \mathbf{n}=0, \mbox{ on } \partial\Omega, \end{cases} \end{eqnarray*} where $\Omega \subset \mathbb{R}^{3}$ is a bounded simply connected domain with a $C^{1,1}$-boundary, denoted by $\partial \Omega$, $p:\overline{\Omega}\to (1, +\infty)$ is a continuous function, $a \in L^\infty(\Omega)$, $f,g : \Omega \times \mathbb{R}^{3}\to \mathbb{R}^{3}$ are Carath\'{e}odory functions, and $\lambda,\mu$ are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang et al. (J. Math. Anal. Appl., 2017), Bahrouni and Repov\v{s} (Complex Var. Elliptic Equ., 2018), and Ge and Lu (Mediterr. J. Math., 2019).

We study the existence and multiplicity results for the following nonlocal $p(x)$-Kirchhoff problem: \begin{equation} \label{10} \begin{cases} -\left(a-b\int_\Omega\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla u)=\lambda |u| ^{p(x)-2}u+g(x,u) \mbox{ in } \Omega, \\ u=0,\mbox{ on } \partial\Omega, \end{cases} \end{equation} where $a\geq b > 0$ are constants, $\Omega\subset \mathbb{R}^N$ is a bounded smooth domain, $p\in C(\overline{\Omega})$ with $N>p(x)>1$, $\lambda$ is a real parameter and $g$ is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.