Periodic perturbations of linear problems at resonance on convex domains (Q1174941)

The paper concerns the semilinear problem \[ \Delta u+\lambda_ 1u+g(u)=h(x),\;x\in\Omega,\quad u=0,\;x\in\partial\Omega, \leqno(1) \] where \(\Omega\subset\mathbb{R}^ n\) is a bounded domain, \(g: \mathbb{R}\to\mathbb{R}\) and \(h: \bar\Omega\to\mathbb{R}\) are Hölder continuous while \(\lambda_ 1\) is the first eigenvalue of the associated (linear) eigenvalue problem; moreover, it is assumed that \(\int_ \Omega h\phi dx=0\), \(g(s+T)=g(s)\) \(\forall s\in\mathbb{R}\), \(\int_ 0^ T g(s)ds=0\), \(g\neq0\), where \(\phi\) is an eigenfunction belonging to \(\lambda_ 1\). The authors prove the existence of infinitely many positive and infinitely many negative solutions to (1). The method of proof relies on embedding (1) into a one-parameter family of problems and then applying bifurcation and continuation techniques.