Resonant problems with multidimensional kernel and periodic nonlinearities. (Q1413579)

The authors study the Dirichlet boundary value problem \[ -u''(t)- u(t)+ A g(u(t))= p(t),\quad t\in [0,\pi],\quad u(0)= u(\pi)= 0, \] where \(g:\mathbb R^n\to\mathbb R^n\) is a given function defined by \(g(u)=(g_1(u_1), g_2(u_2),\dots, g_n(u_n))\), \(\forall u = (u_1, u_2, \dots , u_n)\in\mathbb R^n\) and each \(g_i\), \(i=1,\dots, n\), is a continuous \(T_i\)-periodic function with zero mean value, \(A\) is a constant, regular real \(n\times n\)-matrix and \(p:[0,\pi]\to\mathbb R^n\) is continuous. Existence and multiplicity of solutions to the problem stated above are studied. Infinitely many solutions are obtained under certain assumptions. There are only very few previous works for this kind of problems, see [\textit{A. Cañada}, J. Math. Anal. Appl. 243, 174--189 (2000; Zbl 0952.34013), \textit{P. Drábek} and \textit{S. Invernizzi}, J. Differ. Equations 70, 390--402 (1987; Zbl 0652.34049) and \textit{J. Mawhin}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, Suppl., 415--434 (1989; Zbl 0688.70019)]. The problems under consideration are not necessarily conservative. The methods used in this paper are topological ones; the Lyapunov-Schmidt reduction is accompanied with a careful analysis of oscillatory integrals arising from some bifurcation equation.