Existence of Solutions for the Fractional (p, q)-Laplacian Problems Involving a Critical Sobolev Exponent

Abstract

In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent:

$$({P_{\mu ,\lambda }})\left\{ {\begin{array}{*{20}{l}} {( - \Delta )_p^{{s_1}}u + ( - \Delta )_q^{{s_2}}u = \mu |u{|^{q - 2}}u + \lambda |u{|^{p - 2}}u + |u{|^{p_{{s_1}}^* - 2}}u,}&{\text{in}\;\Omega ,} \\ {u = 0,}&{\text{in}\;{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where Ω ⊂ ℝ^N is a smooth and bounded domain, λ, μ > 0, 0 < S₂ < s₁ < 1, $1 < q < p < {\textstyle{N \over {{s_1}}}}.$ We establish the existence of a non-negative nontrivial weak solution to (P_μ,λ) 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.