This paper deals with the following Kirchhoff–Schrödinger–Newton system with critical growth { − M ( ∫ Ω | ∇ u | 2 d x ) Δ u = ϕ | u | 2 ∗ − 3 u + λ | u | p − 2 u , i n     Ω , − Δ ϕ = | u | 2 ∗ − 1 , i n     Ω , u = ϕ = 0 , o n     ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a smooth bounded domain, M ( t ) = 1 + b t θ − 1 with t > 0 , 1 < θ < N + 2 N − 2 , b > 0 , 1 < p < 2 , λ > 0 is a parameter, 2 ∗ = 2 N N − 2 is the critical Sobolev exponent. By using the variational method and the Brézis–Lieb lemma, the existence and multiplicity of positive solutions are established.

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<abstract><p>Due to its unique performance of high efficiency, fast heating speed and low power consumption, induction heating is widely and commonly used in many applications. In this paper, we study an optimal control problem arising from a metal melting process by using a induction heating method. Metal melting phenomena can be modeled by phase field equations. The aim of optimization is to approximate a desired temperature evolution and melting process. The controlled system is obtained by coupling Maxwell's equations, heat equation and phase field equation. The control variable of the system is the external electric field on the local boundary. The existence and uniqueness of the solution of the controlled system are showed by using Galerkin's method and Leray-Schauder's fixed point theorem. By proving that the control-to-state operator $ P $ is weakly sequentially continuous and Fréchet differentiable, we establish an existence result of optimal control and derive the first-order necessary optimality conditions. This work improves the limitation of the previous control system which only contains heat equation and phase field equation.</p></abstract>