Regularizing effect for a system of Schrödinger–Maxwell equations
Lucio Boccardo, Luigi Orsina · Advances in Calculus of Variations · 2016
Abstract We prove some existence results for the following Schrödinger–Maxwell system of elliptic equations: { - ÷ ( M ( x ) ∇ u ) + A φ | u | r - 2 u = f , u ∈ W 0 1 , 2 ( Ω ) , - ÷ ( M ( x ) ∇ φ ) = | u | r , φ ∈ W 0 1 , 2 ( Ω ) . \left\{\begin{aligned} &\displaystyle{-}\div(M(x) abla u)+A\varphi|u|^{r-2}u=% f,&&\displaystyle u\in W_{0}^{1,2}(\Omega),\\ &\displaystyle{-}\div(M(x) abla\varphi)=|u|^{r},&&\displaystyle\varphi\in W_{% 0}^{1,2}(\Omega).\end{aligned}\right. In particular, we prove the existence of a finite energy solution ( u , φ ) {(u,\varphi)} if r > 2 * {r>2^{*}} and f does not belong to the “dual space” L 2 <