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Fractional $p\&q$ Laplacian problems in $\mathbb{R}^{N}$ with critical growth. (arXiv:1801.10449v2 [math.AP] UPDATED)

来源于:arXiv
We deal with the following nonlinear problem involving fractional $p$ and $q$-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 \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $1<p<q<\frac{N}{s}$, $q^{*}_{s}=\frac{Nq}{N-sq}$, $\lambda>0$, $h>0$ is a bounded function and $f$ is a superlinear continuous function with subcritical growth. By using suitable variational arguments and Concentration-Compactness Lemma, we prove the existence of a nontrivial solution for $\lambda$ sufficiently large. 查看全文>>