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This paper is concerned with the time-harmonic semilinear Maxwell equation: ∇ × (∇ × u) + λu = f(x, u) in Ω with the boundary condition ν × u = 0 on ∂Ω, where Ω ⊂ ℝ3 is a simply connected, smooth, bounded domain with connected boundary and ν : ∂Ω → ℝ3 is the exterior normal. Here ∇ × denotes the curl operator in ℝ3 and the boundary condition holds when Ω is surrounded by a perfect conductor. By using the generalized Nehari manifold method due to Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (International Press, Somerville, 2010), pp. 597–632] and some new techniques, existence of ground state solutions for above equation is established under some generic conditions on f.

In this paper, we study following Kirchhoff type equation: $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ3 by using property of the Pohožaev identity and some delicate analysis.