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 (2)

• A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids
• Existence and nonexistence of solutions for p(x)-curl systems arising in electromagnetism

Cited by in corpus (2)

• 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