Nonlinear resonant periodic problems. (Q2341781)

The paper studies the periodic problem \[ -(|u'(t)|^{p-2}u'(t))'-\mu u''(t)=f(t,u(t)) \;\text{a.e. on\;} [0,b], \;u(0)=u(b),\;u'(0)=u'(b),\tag{1} \] where \(\mu>0\), \(2\leq p<\infty\), and \(f\) is a Carathéodory function, which is \((p-1)\)-linear near \(\pm \infty\). Note that for \(\mu>0\) the differential operator of problem (1) is nonhomogeneous. The authors prove a multiplicity theorem for problem (1), establishing the existence of three nontrivial solutions. Their hypotheses permit double resonance in any spectral interval \([\lambda_m, \lambda_{m+1}]\), \(m\geq 1\). They also work with problems that are resonant with respect to the principal eigenvalue \(\lambda_0\) and establish the existence of nodal solutions. The proofs are based on the critical point theory, coupled with suitable truncation and perturbation techniques and with Morse theory.