Existence of bound state solutions for degenerate singular perturbation problems with sign-changing potentials

In this article, we study the degenerate singular perturbation problems $$displaylines{ -varepsilon^2hbox{div}(|x|^{-2a} abla u)+|x|^{-2(a+1)}V(x)u = |x|^{-b2^*(a,b)}g(x,u),cr -hbox{div}(|x|^{-2a} abla u)+ lambda |x|^{-2(a+1)}V(x)u = |x|^{-b2^*(a,b)}g(x,u), }$$ for $varepsilon$ small and $lambda$ large positive, where $x in mathbb{R}^N$ with $N geq 3$. We search for solutions that decay to zero as $|x| o +infty$, when g is superlinear in the potential function changes signs. We prove the existence of bound state solutions for degenerate, singular, semilinear elliptic problems. Additionally, when the nonlinearity g(x,u) is an odd function of u, we obtain infinitely many geometrically distinct solutions.