论文标题
$ \ mathbb {r}^3 $中两个竞争非线性的非线性麦克斯韦方程的解决方案
Solutions to a nonlinear Maxwell equation with two competing nonlinearities in $\mathbb{R}^3$
论文作者
论文摘要
我们对非线性，时间谐波的麦克斯韦方程$ \ nabla \ times（\ nabla \ times \ times \ times \ mathbf {e}） + v（x）\ mathbf {e} = h（x，x，x，\ mathbf {e}）我们假设$ h $是$$ h（x，αw）= f（x，α）w -g（x，α）w $$ for $ w \ in \ mathbb {r}^3 $，$ | w | = 1 $和$α\ in \ mathbb {r} $。特别是，我们可以考虑由两个竞争力组成的非线性$ h（x，\ mathbf {e}）= | \ Mathbf {e} |^{p-2} \ Mathbf {e} - | \ Mathbf {e} - | \ Mathbf {E}在可靠的假设下，我们表明，特殊形式的弱的，圆柱上的含量溶液是与具有奇异潜力的Schrödinger方程弱解的一对一对应关系。使用这种等价结果，我们显示了特定形式的圆柱式溶液与麦克斯韦方程以及schrödinger方程的圆柱溶解解决方案之间存在最少的能量解。
We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$ h(x, αw) = f(x, α) w - g(x, α) w $$ for $w \in \mathbb{R}^3$, $|w|=1$ and $α\in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of the special form are in one-to-one correspondence with weak solutions to a Schrödinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant solutions of the particular form to the Maxwell equation, as well as to the Schrödinger equation.
