On the Fredholm alternative for the Fučík spectrum (Q624460)

Let \(p>1\), \(T>0\), \((\alpha,\beta)\in\mathbb R^2\), and \(f\in L^1(0,T)\). This paper is concerned with the qualitative analysis of the quasilinear problem \[ -\left( |u'|^{p-2}u'\right)'-\alpha (u^+)^{p-1}+ \beta (u^-)^{p-1}=f\quad\text{in}\;(0,T), \] subject to the Dirichlet boundary condition \(u(0)=u(T)=0\). The authors are interested in the resonant case that corresponds to \((\alpha,\beta)\in\Sigma_p\), where \(\Sigma_p\) denotes the Fučík spectrum associated to the above problem. By means of variational arguments based on linking theorems, the authors establish sufficient conditions for the existence of at least one solution. The main results in the present paper are related to the classical Fredholm alternative for linear operators. The authors also provide a new variational characterization for points on the third Fučík curve.