Superlinear variational and boundary value problems with parameters
The paper is devoted to the question of the continuous dependence on parameters \(w\) of the set of critical points of the functional \[ F_w(u)= \int_\Omega \Biggl[{1\over 2}|\nabla u(x)|^2- G(x,u(x), w(x))\Biggr] dx \] defined on \(H^1_0(\Omega)\) and the set of solutions of the associated Euler-Lagrange system \[ \Delta u(x)+ \nabla_u G(x,u(x), w(x)) \] under the assumption of superlinearity \[ a< pG(x,u,w)\leq \langle\nabla_u G(x,u,w), u\rangle \] for some \(a> 0\) and \(p> 2\). Some special attention is paid to the one-dimensional problem.
scientific article; zbMATH DE number 1545350
scientific article