Existence via regularity of solutions for elliptic systems and saddle points of functionals of the calculus of variations
Lucio Boccardo, Luigi Orsina · Advances in Nonlinear Analysis · 2016
Abstract The core of this paper concerns the existence (via regularity) of weak solutions in W 0 1 , 2 ${W_{0}^{1,2}}$ of a class of elliptic systems such as { - div ( ( A + φ ) ∇ u ) = f , - div ( M ( x ) ∇ φ ) = 1 2 | ∇ u | 2 , $\left\{\begin{aligned} \displaystyle-\operatorname{div}((A+\varphi) abla u)&% \displaystyle=f,\\ \displaystyle-\operatorname{div}(M(x) abla\varphi)&\displaystyle=\frac{1}{2}% \lvert abla u\rvert^{2},\end{aligned}\right.$ deriving from saddle points of integral functionals of the type J ( v , ψ ) = 1 2 ∫ Ω ( A + ψ + ) | ∇ v | 2