In this talk, we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains of the form . Using weak -coercivity theory, we can establish the existence of an orthonormal basis of eigenfunctions of the linear part
-c(x)^-1div(σ(x)∇u). Then, all eigenvalues are proved to be bifurcation points and we investigate the bifurcating branches both theoretically and numerically. As a fundamental example, we look at some one-dimensional model, we obtain the existence of infinitely many bifurcating branches that are mutually disjoint, unbounded, and consist of solutions with a fixed nodal pattern.