Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media

Abstract

In this work we obtain an existence result for a class of singular quasilinearelliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenbergtype inequality, a critical point result fordifferentiable functionals is exploited, in order to prove theexistence of a precise open interval of positive eigenvalues forwhich the treated problem admits at least one nontrivial weak solution. In the case ofterms with a sublinear growth near the origin, we deduce the existence of solutions for small positive values of the parameter. Moreover, the corresponding solutions have smaller and smaller energies as the parameter goes to zero.