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Zeitschrift für Analysis und ihre Anwendungen


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Volume 39, Issue 3, 2020, pp. 289–314
DOI: 10.4171/ZAA/1661

Published online: 2020-07-06

Fractional $p\&q$ Laplacian Problems in $\mathbb{R}^{N}$ with Critical Growth

Vincenzo Ambrosio[1]

(1) Università Politecnica delle Marche, Ancona, Italy

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.

Keywords: fractional $p\&q$ Laplacians, variational methods, critical exponent

Ambrosio Vincenzo: Fractional $p\&q$ Laplacian Problems in $\mathbb{R}^{N}$ with Critical Growth. Z. Anal. Anwend. 39 (2020), 289-314. doi: 10.4171/ZAA/1661