Nonnegative solutions to a weighted indefinite elliptic Schrödinger equation

In this paper, we consider the following weighted problem involving an indefinite nonlinearity: $${aligned} \{ \! {array}{lllllll} -div(g(x)∇ u)\!=\!λ ₁β (x)u\!+\!μ f(x,u)\!+\!W(x)h(u)\!+\! η u2_s^*\!-\!1\ in \ Ω ,\\ u=0 \ on \ ∂ Ω ,\\ u(x)≥ 0 \ in \ Ω . {array}. (Pμ ,η) {aligned}$$ - div ( g ( x ) ∇ u ) = λ 1 β ( x ) u + μ f ( x , u ) + W ( x ) h ( u ) + η u 2 s ∗ - 1 in Ω , u = 0 on ∂ Ω , u ( x ) ≥ 0 in Ω . ( P μ , η ) where <jats:alternati