Fractional (p, q)-Schrodinger Equations with Critical and Supercritical Growth

In this paper, we complete the study started in Ambrosio and Radulescu (J Math Pures Appl (9) 142:101-145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrodinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional ( p, q)-Laplacian problems: {(-Delta)(p)(s)u + (-Delta)(q)(s)u + V (epsilon x)(u(p-1) + u(q-1)) = f(u) + u(qs*-1) in R-N, u is an element of W-s,W-p (R-N) boolean AND W-s,W-q (R-N), u > 0 in R-N. where epsilon > 0 is a small parameter, s is an element of (0, 1), 1 < p < q < N/s, q(s)* = Nq/N-sq is the fractional critical Sobolev exponent, (-Delta)(r)(s), with r is an element of {p, q}, is the fractional r-Laplacian operator, V : R-N -> R is a positive continuous potential such that inf(partial derivative Lambda) V > inf(Lambda) V for some bounded open set Lambda subset of R-N, and f : R -> R is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik-Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when f (t) = t(gamma-1) + mu(t tau-1), with 1 < p < q < gamma < q(s)* < t and mu > 0 sufficiently small, by combining a truncation argument with a Moser-type iteration.