The multiplicity of positive solutions to the quasilinear equation generated by the Il'in–Caffarelli–Kohn–Nirenberg inequality

We consider the Euler–Lagrange equation for the functional related to the V. P. Il'in inequality also known as the Caffarelli–Kohn–Nirenberg inequality. We prove that if the space dimension is even then, changing some parameters, we can obtain arbitrary many different positive solutions for this equation.