A class of resonant or indefinite elliptic problems (Q2711962)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N(N\geq 3)\) with smooth boundary. The author deals with the elliptic problem NEWLINE\[NEWLINE\begin{cases} -\Delta u= \lambda u-a(x)|u|^{q-1} \cdot u+f(x,y) \text{ in }\Omega,\\ u |_{ \partial \Omega}= 0\end{cases} \tag{1}NEWLINE\]NEWLINE where \(a(x)\in L^\infty(\Omega)\), \(0<q<1\), and the function \(f\) is sublinear at infinity, while \(\lambda\) is a parameter. The author shows the existence and uniqueness of a positive solution with the help of a monotone argument combined with a minimizing procedure. Moreover, the existence of infinitely many nontrivial solutions are obtained by the oddness of the nonlinear term according to the index theory.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00021].