一类非线性常系数curl-curl方程柱对称基态的存在性
Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients
摘要：
考虑了$\mathbb{R}^3$中的非线性卷曲-卷曲问题$\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$，该问题与具有Kerr-type非线性物质定律的非线性Maxwell方程组有关.我们证明了一个有界的，圆柱对称系数$V$和亚临界圆柱对称非线性$f$的对称基态解的存在性。新的存在性结果扩展了一类问题的基态类型的解决方案是已知的。它是基于紧性的对称函数由于狮子，新的重排型不等式布洛克和最近的扩展Nehari流形技术由Szulkin和Weth。
英文摘要：
We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient $V$ and subcritical cylindrically symmetric nonlinearity $f$. The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions, new rearrangement type inequalities from Brock and the recent extension of the Nehari-manifold technique by Szulkin and Weth.
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