Large solution of a semilinear elliptic problem

AbstractWe show that large positive solutions exist for the following equationΔu+|∇u|q=p(x)f(u)(p+)in Ω ⊆ RN (N ≥ 3) in which the domain Ω is either bounded or equal to RN. The nonnegative function p is continuous and may vanish on large parts of Ω. If Ω = RN, then p must satisfy a decay condition∫0∞rϕ(r)dr<∞,whereϕ(r)=max⁡|x|=rp(x)as|x|→∞.Furthermore, we show that the given conditions on p are nearly optimal for equation (p+)

Lifshitz Scaling Effects on Holographic Superconductors

Via numerical and analytical methods, the effects of the Lifshitz dynamical exponent $z$ on holographic superconductors are studied in some detail, including $s$ wave and $p$ wave models. Working in the probe limit, we find that the behaviors of holographic models indeed depend on concrete value of $z$. We obtain the condensation and conductivity in both Lifshitz black hole and soliton backgrounds with general $z$. For both $s$ wave and $p$ wave models in the black hole backgrounds, as $z$ increases, the phase transition becomes more difficult and the growth of conductivity is suppressed. For the Lifshitz soliton backgrounds, when $z$ increases ($z=1,~2,~3$), the critical chemical potential decreases in the $s$ wave cases but increases in the $p$ wave cases. For $p$ wave models in both Lifshitz black hole and soliton backgrounds, the anisotropy between the AC conductivity in different spatial directions is suppressed when $z$ increases. The analytical results uphold the numerical results.Comment: Typos corrected; Footnote added; References added; To be published in Nuclear Physics

On-chip electro-optic tuning of a lithium niobate microresonator with integrated in-plane microelectrodes

We demonstrate electro-optic tuning of an on-chip lithium niobate microresonator with integrated in-plane microelectrodes. First two metallic microelectrodes on the substrate were formed via femtosecond laser process. Then a high-Q lithium niobate microresonator located between the microelectrodes was fabricated by femtosecond laser direct writing accompanied by focused ion beam milling. Due to the efficient structure designing, high electro-optical tuning coefficient of 3.41 pm/V was observed.Comment: 6 pages, 3 figure