A critical fractional equation with concave–convex power nonlinearities
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, p. 875-900
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In this work we study the following fractional critical problem $\left({P}_{\lambda }\right)=\left\{\begin{array}{cc}{\left(-\Delta \right)}^{s}u=\lambda {u}^{q}+{u}^{{2}_{s}^{⁎}-1},\phantom{\rule{1em}{0ex}}u>0\hfill & \text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\hfill \\ u=0\hfill & \text{in}\phantom{\rule{4pt}{0ex}}{ℝ}^{n}\setminus \Omega ,\hfill \end{array}$ where $\Omega \subset {ℝ}^{n}$ is a regular bounded domain, $\lambda >0$, $0 and $n>2s$. Here ${\left(-\Delta \right)}^{s}$ denotes the fractional Laplace operator defined, up to a normalization factor, by $-{\left(-\Delta \right)}^{s}u\left(x\right)=\underset{{ℝ}^{n}}{\int }\frac{u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)}{{|y|}^{n+2s}}\phantom{\rule{0.166667em}{0ex}}dy,\phantom{\rule{1em}{0ex}}x\in {ℝ}^{n}.$ Our main results show the existence and multiplicity of solutions to problem $\left({P}_{\lambda }\right)$ for different values of λ. The dependency on this parameter changes according to whether we consider the concave power case ($0) or the convex power case ($1). These two cases will be treated separately.

DOI : https://doi.org/10.1016/j.anihpc.2014.04.003
Classification:  49J35,  35A15,  35S15,  47G20,  45G05
Keywords: Fractional Laplacian, Critical nonlinearities, Convex–concave nonlinearities, Variational techniques, Mountain Pass Theorem
@article{AIHPC_2015__32_4_875_0,
author = {Barrios, B. and Colorado, E. and Servadei, R. and Soria, F.},
title = {A critical fractional equation with concave--convex power nonlinearities},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {4},
year = {2015},
pages = {875-900},
doi = {10.1016/j.anihpc.2014.04.003},
zbl = {1350.49009},
mrnumber = {3390088},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_4_875_0}
}

Barrios, B.; Colorado, E.; Servadei, R.; Soria, F. A critical fractional equation with concave–convex power nonlinearities. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 875-900. doi : 10.1016/j.anihpc.2014.04.003. http://www.numdam.org/item/AIHPC_2015__32_4_875_0/

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