On indefinite sublinear elliptic equations (Q2721905)

Let \(\Omega\subset \mathbb{R}^d\), \(d\geq 3\) be a smooth bounded domain and consider the semilinear elliptic equations NEWLINE\[NEWLINE\begin{cases} -\Delta u=\lambda u+a(x) u^q-b(x) u^p,\;u>0\quad &\text{in }\Omega\\ u=0\quad \quad & \text{on }\partial \Omega,\end{cases} \tag{1}NEWLINE\]NEWLINE where \(a,b\in L^\infty(\Omega)\), \(\lambda \in\mathbb{R}^1\) is a parameter. The authors study the solution of (1) in the indefinite sublinear case, that is \(0<q<p<1\) and \(b(x)\) many change sign. Under some natural conditions on \(a\) and \(b\), the authors prove existence of at least two selections in the right neighborhood of the first eigenvalue of \(-\Delta\) with zero Dirichlet condition. To this end they use variational and sub-sup-solution methods.