Using the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences: {div⁡(a⁢(x,∇⁡u))=0in ⁢Ω,a⁢(x,∇⁡u)⋅ν=λ⁢m⁢(x)⁢|u|p⁢(x)-2⁢uon ⁢∂⁡Ω,\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}}(N≥2){(N\geq 2)} is a bounded domain of smooth boundary ∂⁡Ω{\partial\Omega} and ν is the outward unit normal vector on ∂⁡Ω{\partial\Omega}. The functions m∈L∞⁢(∂⁡Ω){m\in L^{\infty}(\partial\Omega)}, p:Ω¯↦ℝ{p\colon\overline{\Omega}\mapsto\mathbb{R}} and a:Ω¯×ℝN↦ℝN{a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

中文翻译：

---

具有 a 谐波解和可变指数的 Steklov 特征值问题

使用 Ljusternik-Schnirelmann 原理和一种新的变分技术，我们证明了以下 Steklov 特征值问题具有无限多个正特征值序列：{div⁡(a⁢(x,∇⁡u))=0in ⁢Ω,a⁢(x ,∇⁡u)⋅ν=λ⁢m⁢(x)⁢|u|p⁢(x)-2⁢uon ⁢∂⁡Ω,\left\{\begin{aligned} &\displaystyle\operatorname{div} (a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x )|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. 其中 Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}}(N≥2){(N\geq 2)} 是平滑边界 ∂⁡Ω{\partial\Omega} 和 ν 的有界域是∂⁡Ω{\partial\Omega} 上的外向单位法向量。函数 m∈L∞⁢(∂⁡Ω){m\in L^{\infty}(\partial\Omega)}, p:Ω¯↦ℝ{p\colon\overline{\Omega}\mapsto\mathbb {R}} 和一个：