Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents

In this paper, we consider the equation−Δpu=λ|u|p⁎−2u+f(x,u)in RN, with discontinuous nonlinearity, where 10λ>0 is a real parameter and p⁎=NpN−p is the critical Sobolev exponent. Under proper conditions on f  , applying the nonsmooth critical point theory for locally Lipschitz functionals, we obtain at least one nontrivial nonnegative solution provided that λ<λ0λ<λ0 and for any k∈Nk∈N, it has k   pairs of nontrivial solutions if λ<λkλ<λk, where λ0λ0 and λkλk are positive numbers. In particular, we obtain the existence results for f is discontinuous in just one point.