On linear and nonlinear fourth-order eigenvalue problems with indefinite weight

We determine the principal eigenvalues of the linear indefinite weight problem u⁗(x)=rg(x)u(x),x∈(0,1),u(0)=u(1)=u″(0)=u″(1)=0. Moreover, we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem, where g:[0,1]→Rg:[0,1]→R is a continuous function which attains both positive and negative values, f∈C(R,R)f∈C(R,R), and rr is a parameter.