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

In this paper, we consider the equation − Δ p u = λ | u | p ⁎ − 2 u + f ( x , u ) in R N , with discontinuous nonlinearity, where 1 < p < N , λ > 0 is a real parameter and p ⁎ = N p N − 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 < λ 0 and for any k ∈ N , it has k pairs of nontrivial solutions if λ < λ k , where λ 0 and λ k are positive numbers. In particular, we obtain the existence results for f is discontinuous in just one point.