High-Qf value and temperature stable Zn2+-Mn4+ cooperated modified cordierite-based microwave and millimeter-wave dielectric ceramics

Cordierite-based dielectric ceramics with a lower dielectric constant would have significant application potential as dielectric resonator and filter materials for future ultra-low-latency 5G/6G millimeter-wave and terahertz communication. In this article, the phase structure, microstructure and microwave dielectric properties of Mg2Al4–2x(Mn0.5Zn0.5)2xSi5O18 (0 ≤ x ≤ 0.3) ceramics are studied by crystal structure refinement, scanning electron microscope (SEM), the theory of complex chemical bonds and infrared reflectance spectrum. Meanwhile, complex double-ions coordinated substitution and two-phase complex methods were used to improve its Q×f value and adjust its temperature coefficient. The Q×f values of Mg2Al4–2x(Mn0.5Zn0.5)2xSi5O18 single-phase ceramics are increased from 45,000 [email protected] GHz (x = 0) to 150,500 [email protected] GHz (x = 0.15) by replacing Al3+ with Zn2+-Mn4+. The positive frequency temperature coefficient additive TiO2 is used to prepare the temperature stable Mg2Al3.7(Mn0.5Zn0.5)0.3Si5O18-ywt%TiO2 composite ceramic. The composite ceramic of Mg2Al3.7(Mn0.5Zn0.5)0.3Si5O18-ywt%TiO2 (8.7 wt% ≤ y ≤ 10.6 wt%) presents the near-zero frequency temperature coefficient at 1225 °C sintering temperature: εr = 5.68, Q×f = 58,040 GHz, τf = −3.1 ppm/°C (y = 8.7 wt%) and εr = 5.82, Q×f = 47,020 GHz, τf = +2.4 ppm/°C (y = 10.6 wt%). These findings demonstrate promising application prospects for 5 G and future microwave and millimeter-wave wireless communication technologies

p-Kirchhoff type problem with a general critical nonlinearity

In this article, we consider the p-Kirchhoff type problem $$\Big(1+\lambda\int_{\mathbb{R}^N}|\nabla u|^p +\lambda b\int_{\mathbb{R}^N}|u|^p\Big)(-\Delta_p u+b|u|^{p-2}u) =f(u), x\in\mathbb{R}^N,$$ where $\lambda>0$, the nonlinearity f can reach critical growth. Without the Ambrosetti-Robinowitz condition or the monotonicity condition on f, we prove the existence of positive solutions for the p-Kirchhoff type problem. In addition, we also study the asymptotic behavior of the solutions with respect to the parameter $\lambda\to0$

p-Kirchhoff type problem with a general critical nonlinearity

In this article, we consider the p-Kirchhoff type problem (1 + λ ∫ℝN |∇u|p + λb ∫ℝN |u|p) (-∆pu + b|u|p-2u) = ƒ(u), x ∈ ℝN, where λ > 0, the nonlinearity ƒ can reach critical growth. Without the Ambrosetti-Robinowitz condition or the monotonicity condition on ƒ, we prove the existence of positive solutions for the p-Kirchhoff type problem. In addition, we also study the asymptotic behavior of the solutions with respect to the parameter λ → 0.Mathematic