Combined effects of concave and convex nonlinearities in some elliptic problems (Q1328326)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ N\) and consider the semilinear elliptic problem \[ -\Delta u = f_ \lambda (x,u) \quad \text{ in } \Omega, \quad u = 0 \quad \text{ on } \partial \Omega \tag{1} \] where \(f_ \lambda : \Omega \times \mathbb{R} \to \mathbb{R}\) and \(\lambda\) is a real parameter. When \(f_ \lambda\) is sublinear, for example, \(f_ \lambda = \lambda u^ q\), \(0<q<1\), it is known that (1) has a unique positive solution for all \(\lambda>0\). On the other hand, if \(f_ \lambda = \lambda | u |^{q-1}u\), the problem (1) admits infinitely many solutions. The purpose of the present note is to study (1) when \(f_ \lambda\) is the sum of a sublinear and superlinear term. It is shown that the combined effects of these two nonlinearities change considerably the structure of the solution set.