EXISTENCE OF A WEAK SOLUTION FOR DISCONTINUOUS ELLIPTIC PROBLEMS INVOLVING THE FRACTIONAL p(.)-LAPLACIAN

We are concerned with the following nonlinear elliptic equation of the fractional p(.)-Laplace type: {(-Delta(s)(p(x))u + vertical bar u vertical bar(p(x)-2)u is an element of lambda[(g) under bar (x, u(x), (g) over bar (x, u(x))] in Omega, u=0 on R-N\Omega where (-Delta)(p(.))(s) is the fractional p(.)-Laplacian operator, lambda is a parameter, 0 < s < 1, sp(+) < N, and the measurable functions <(g)under bar>, (g) over bar are induced by a possibly discontinuous function g : Omega x R -> R at the second variable. By using the Berkovits-Tienari degree theory for upper semicontinuous set-valued operators of type (S+) in reflexive Banach spaces, we show that our problem with the discontinuous nonlinearity g possesses at least one nontrivial weak solution.