In this article, we study the following fractional $(p,q)$-Laplacian equations involving the critical Sobolev exponent: \[ (P_{\mu, \lambda}) \begin{cases} (-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u=\mu |u|^{q-2}u +\lambda|u|^{p-2}u + |u|^{p_{s_{1}}^{*}-2}u, & \text{in $\Omega$,} \\ u=0, & \text{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, $\lambda,\ \mu >0, \ 0 < s_{2} < s_{1} < 1,\ 1 < q < p < \frac{N}{s_{1}} $. We establish the existence of a non-negative nontrivial weak solution to $(P_{\mu, \lambda})$ 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.
Fanfan CHEN
,
Yang YANG
. EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1666
-1678
.
DOI: 10.1007/s10473-020-0604-9
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