Multiple solutions to some degenerate quasilinear equation with variable exponents via perturbation method

In this paper, we prove the existence of signed and sign-changing solutions to some degenerate quasilinear and elliptic equation containing a gradient term and involving a variable exponent. Our main idea is to make use of the perturbation method together with invariant sets of descending flow. The fact that the exponent is not constant plays an important role.