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 { (P) δ } − ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i − 2 ∂ u ∂ x i = ∫ Ω F (x , u) r f (x , u) + δ | u | p ∗ − 2 u in Ω , u ≥ 0 in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain of R N , N ≥ 2 , δ = 0 or δ = 1 , p i and r ≥ 0 , 1 < p 1 ≤ ⋯ ≤ p N < p ∗ are parameters where p ∗ = N p ¯ / (N − p ¯) is a critical exponent, p ¯ = N / ∑ i = 1 N 1 / p i and p ¯ < N. The nonlinearity f : Ω × R → R can be discontinuous, has subcritical growth subcritical growth and F (x , t) = ∫ 0 t f (x , s) d s. Under appropriate assumptions on f, we obtain the existence of nontrivial solutions for (P) δ using the Ekeland's Variational Principle, Nonsmooth Mountain Pass Theorem and a Concentration Compactness-Principle. [ABSTRACT FROM AUTHOR]