Representation theorems for Sobolev spaces on intervals and multiplicity results for nonlinear ODEs (Q615970)

The authors present first topological isomorphisms between \(W^{m,p}(0,1)\) and \(L^p(0,1)\times\mathbb{R}^m\) for \(m\geq1\) and \(1\leq p\leq\infty.\) The main result of the paper is to prove the existence of infinitely many classical solutions to the second-order nonlinear, nonhomogeneous differential system \[ -u''=| v|^{p-1}v+f(x),\quad x\in(0,1) \] \[ -v''=| u|^{q-1}u+g(x),\quad x\in(0,1) \] subject to the boundary conditions \[ u,v=0\;\text{ on }\;\{0,1\}, \] where \(f, g\in C^1[0,1]\) and \(p, q>0\), \(pq>1\). The authors use arguments from Fourier analysis and the study of Schauder bases for \(W^{m,p}(0,1)\). Applications to a perturbed symmetric system and a fourth-order concave-convex quasilinear ODE are also given.