Multiple solutions for p(x)${p(x)}$‐curl systems with nonlinear boundary condition

Abstract

In this paper, we study the 𝑝(𝑥)-curl problem of the type:where Ω ⊂ ℝ 3 is a bounded simply connected domain with a 𝐶 1,1 boundary denoted by 𝜕Ω, 𝜆 > 0, 𝑝(𝑥) ∶ Ω → (1, +∞) is a continuous function, 𝑎(𝑥) ∈ 𝐿 ∞ (Ω), and 𝐟 ∶ Ω × ℝ 3 → ℝ 3 is a Carathéodory function. We obtain the existence and multiplicity of solutions for a class of 𝑝(𝑥)-curl systems in the absence of Ambrosetti-Rabinowitz condition under superlinear case. Besides, we also obtain infinitely many solutions under sublinear case.