Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2017-10-09
Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant CoefficientsAndreas Hirsch and Wolfgang Reichel (1) Karlsruhe Institute of Technology (KIT), Germany
(2) Karlsruhe Institute of Technology (KIT), Germany
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 [J. Funct. Anal. 41 (1981)(2), 236–275], new rearrangement type inequalities by Brock [Proc. Indian Acad. Sci. Math. Sci. 110 (2000), 157–204] and the recent extension of the Nehari-manifold technique from Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (2010), pp. 597–632].
Keywords: Curl-curl problem, nonlinear elliptic equations, cylindrical symmetry, variational methods
Hirsch Andreas, Reichel Wolfgang: Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients. Z. Anal. Anwend. 36 (2017), 419-435. doi: 10.4171/ZAA/1595