In this work, we study the existence and multiplicity results for the following nonlocal p(x)-Kirchhoff problem:

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

In this paper, we study the following p(x)-curl systems: ∇ × (|∇ × u| p(x)−2 ∇ × u) + a(x)|u| p(x)−2 u = λf (x, u) + µg(x, u), ∇ • u = 0, in Ω, |∇ × u| p(x)−2 ∇ × u × n = 0, u • n = 0, on ∂Ω, where Ω ⊂ R 3 is a bounded simply connected domain with a C 1,1-boundary, denoted by ∂Ω, p : Ω → (1, +∞) is a continuous function, a ∈ L ∞ (Ω), f, g : Ω × R 3 → R 3 are Carathéodory functions, and λ, µ 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.

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In this paper, we prove the existence of multiple solutions for the following sixth-order p(x)-Kirchhoff-type problem − M ∫ Ω 1 p ( x ) | ∇ Δ u | p ( x ) d x Δ p ( x ) 3 u = λ f ( x ) | u | q ( x ) − 2 u + g ( x ) | u | r ( x ) − 2 u + h ( x ) in Ω , u = Δ u = Δ 2 u = 0 , on ∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -M\left( \int\limits_{\it\Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\[0.3em] u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ N is a smooth bounded domain, N > 3 , Δ p ( x ) 3 u := div ⁡ ( Δ ( | ∇ Δ u | p ( x ) − 2 ∇ Δ u ) ) $\begin{array}{} N \,\,\gt\,\, 3, {\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big) \end{array}$ is the p(x)-triharmonic operator, p, q, r ∈ C( Ω ), 1 < p(x) < N 3 $\begin{array}{} \displaystyle \frac N3 \end{array}$ for all x ∈ Ω , M(s) = a − bsγ , a, b,γ > 0, λ > 0, g : Ω × ℝ → ℝ is a nonnegative continuous function while f, h : Ω × ℝ → ℝ are sign-changing continuous functions in Ω. 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.