In this paper, we consider the following weighted problem involving an indefinite nonlinearity: \(\begin{aligned} \left\{ \! \begin{array}{lllllll} -\operatorname {div}(g(x)\nabla u)\!=\!\lambda _{1}\beta (x)u\!+\!\mu f(x,u)\!+\!W(x)h(u)\!+\! \eta u^{2_s^*\!-\!1}\ \text {in} \ \Omega ,\\ u=0 \ \text {on} \ \partial \Omega ,\\ u(x)\ge 0 \ \text {in} \ \Omega . \end{array}\right. \quad (P_{\mu ,\eta }) \end{aligned}\) where \(\Omega \subset \mathbb {R}^{N}\) is a bounded domain with smooth boundary, \(\eta \in \{0,1\}\) , \(N\ge 3\) , \(\lambda _{1}\) denotes the first eigenvalue of the corresponding problem in \(\Omega \) , \(g^{-1}\in L^{1}_{loc}(\Omega )\) is a positive weight, and \(\beta \in L^\infty (\Omega )\) . The function W is indefinite and changes sign, and we impose suitable conditions on f and h. The constant \(2_s^*\) is the critical exponent associated with the operator. Applying the mountain pass theorem, we prove the existence of a nonnegative solution for \((P_{\mu ,\eta })\) and, moreover, obtain a pointwise lower bound for the first eigenvalue associated with the operator on an adequate domain.
