Existence of solutions for systems arising in electromagnetism

Abstract

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}=λf(x,\mathbf{u})+μg(x,\mathbf{u}),\quad\nabla\cdot \mathbf{u}=0,\; \mbox{ in } Ω, \\ |\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u}\times \mathbf{n}=0,\quad \mathbf{u}\cdot \mathbf{n}=0, \mbox{ on } \partialΩ, \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 Carathéodory 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 Repovš (Complex Var. Elliptic Equ., 2018), and Ge and Lu (Mediterr. J. Math., 2019).

References in corpus (5)

• A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids
• Stationary waves of Schrödinger-type equations with variable exponent
• Existence and multiplicity results for a new $p(x)$-Kirchhoff problem
• Existence and nonexistence of solutions for p(x)-curl systems arising in electromagnetism
• High-order Kirchhoff problems in bounded and unbounded domains

Cited by in corpus (3)

• New class of sixth-order nonhomogeneous $p(x)$-Kirchhoff problems with sign-changing weight functions
• Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition
• Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms