Continuous dependence on parameters and boundary data for nonlinear P.D.E. coercive case. (English) Zbl 1042.35004

The paper deals with the system of second-order quasilinear elliptic equations being the Euler-Lagrange equation for the functional \(F_u(v)=\int _{\Omega } f(x,v,\nabla v,u)\,dx\) with \(v\in H^1_0 (\Omega )\) and \(u\in L^{\infty }(\Omega )\) a parameter. Convexity and quadratic growth of \(f(x,v,\cdot ,u)\) are assumed. The upper semicontinuity (in Kuratowski’s sense) of the set-valued mapping \(u\mapsto V_u\), the set of minimizers of \(F_u\), is shown via a uniform convergence of the functionals \(F_u\) on bounded sets. The strong \(L^{\infty }\)-topology is used, while in the special case \(f(x,v,\nabla v,u) = f_1 (x,v,\nabla v) + f_2 (x,v) \cdot u\) the weak \(^*\)-topology is sufficient. If \(F_u\) is convex, it is interpreted as well-posedness of the mentioned Euler-Lagrange equation. The results seem, however, of a rather simple character, in particular with respect to the broadly developed theory of \(\Gamma \)-convergence, the connections to which are not mentioned at all.


35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J60 Nonlinear elliptic equations