EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT

In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent: (P-mu,P-lambda){(-Delta)(p)(s1)(-Delta)(q)(s2) u = mu vertical bar u vertical bar(q-2) u + lambda vertical bar u vertical bar(p-2) u + vertical bar u vertical bar(p)*(s1) (- 2) u, in Omega, u = 0, in R-N \ Omega, where Omega subset of R-N is a smooth and bounded domain, lambda, mu > 0, 0 < s(2)<s(1)< 1, 1 < q < p < N/s(1). We establish the existence of a non-negative nontrivial weak solution to (P-mu,P-lambda) by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.