Existence results for an anisotropic nonlocal problem involving critical and discontinuous nonlinearities

In this paper, we are interested in the existence of solutions to the anisotropic nonlocal problem -Sigma(N)(i=1) partial derivative/partial derivative X-i )vertical bar partial derivative u/partial derivative X-i vertical bar(pi-2) partial derivative u/partial derivative X-i) = (integral(Omega) F(x, u))(r) f(x, u) + delta vertical bar u vertical bar(p*-2) u in Omega, u >= 0 in Omega, u = 0 on partial derivative Omega, (P)(delta) where Omega is a smooth bounded domain of R-N, N >= 2, delta = 0 or delta = 1,p(i) and r >= 0, 1 p(1) <= center dot center dot center dot p(N) < p* are parameters where p* = N<(p)over bar>/(N - (p) over bar) is a critical exponent, (p) over bar = N/Sigma(N)(i=1) 1/p(i) and (p) over bar < N. The nonlinearity f : Omega x R -> R can be discontinuous, has subcritical growth subcritical growth and F(x,t) = integral(t)(0) f(x,s) ds. Under appropriate assumptions on f, we obtain the existence of nontrivial solutions for (P)(delta) using the Ekeland's Variational Principle, Nonsmooth Mountain Pass Theorem and a Concentration Compactness-Principle.