We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $1 < p < q < \frac{N}{s}$, $\q=\frac{Nq}{N-sq}$, $\lambda > 0$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for $\lambda$ sufficiently large.

中文翻译：

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具有临界增长的$ \ mathbb {R} ^ {N} $中的分数$ p \＆q $拉普拉斯问题

我们处理以下涉及分数小数$ p \＆q $拉普拉斯算子的非线性问题：\ begin {equation *}（-\ Delta）^ {s} _ {p} u +（-\ Delta）^ {s} _ {q} u + | u | ^ {p-2} u + | u | ^ {q-2} u = \ lambda h（x）f（u）+ | u | ^ {q ^ {*} _ {s} -2} u \ quad \ mbox {in} \ mathbb {R} ^ {N}，\ end {equation *}其中$ s \ in（0,1）$，$ 1 <p <q <\ frac {N} {s} $ ，$ \ q = \ frac {Nq} {N-sq} $，$ \ lambda> 0 $是一个参数，$ h $是一个非平凡的有界摄动，$ f $是具有次临界增长的超线性连续函数。使用适当的变分参数和浓度紧致引理，我们证明了$λ足够大的非平凡非负解的存在。