Nonlinear elliptic systems with variable boundary data
The authors consider systems of elliptic nonlinear partial differential equations \[ -\Delta u(x)=G_u(x,u(x),v(x)), \] \[ \Delta v(x)=G_v(x,u(x),v(x)), \] \[ u=g \text{ on } \Gamma, \quad u=h \text{ on } \Gamma, \] where \(g(\cdot)\in H^{1/2}(\Gamma, \mathbb R^N)\), \(h(\cdot)\in H^{1/2}(\Gamma,\mathbb R^M)\), \(u(\cdot)\in H^{1}(\Omega, \mathbb R^N)\), \(v(\cdot)\in H^{1}(\Omega, \mathbb R^M)\), \(G\) is a scalar function defined on the set \(\Omega\times \mathbb R^N\times \mathbb R^M\), \(G_u=(G_{u^1}),\dots,G_{u^N})\), \(G_v=(G_{v^1}),\dots,G_{v^M})\), \(\Omega\subset \mathbb R^n\), \(n\geq2\), \(\Gamma=\partial \Omega\). The question of the continuous dependence on boundary data \((g,h)\in W\) of solutions (in the weak sense) in the space \(H^1(\Omega, \mathbb R^N)\times H^1(\Omega, \mathbb R^M)\) is investigated. Under some assumption this question is replaced with the question of the continuous dependence on the boundary data of saddle points of the functional of action \[ F(u,v)=\int_{\Omega} \left(\frac{1}{2}\nabla v(x)^2)-\frac{1}{2}\nabla u(x)^2 +G(x,u(x),v(x))\right) dx \] for the considered system.
scientific article; zbMATH DE number 1880012
scientific article