A critical fractional equation with concave–convex power nonlinearities

Barrios, B.; Colorado, E.; Servadei, R.; Soria, F.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 875-900.

In this work we study the following fractional critical problem

$$\left(P_{\lambda }\right)=\{\begin{array}{cc}{(-\Delta )}^{s}u=\lambda u^q+u^{2_s^{⁎}-1},u>0\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{in}{ℝ}^n\setminus \Omega ,\hfill \end{array}$$

where $\Omega \subset {ℝ}^n$ is a regular bounded domain, $\lambda >0$, $0<s<1$ and $n>2s$. Here ${(-\Delta )}^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by

$$-{(-\Delta )}^{s}u\left(x\right)=\underset{{ℝ}^n}{\int }\frac{u(x+y)+u(x-y)-2u\left(x\right)}{{\left|y\right|}^{n+2s}}dy,x\in {ℝ}^n.$$

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 ($0<q<1$) or the convex power case ($1<q<2_s^{⁎}-1$). These two cases will be treated separately.

Classification : 49J35, 35A15, 35S15, 47G20, 45G05
Mots clés : Fractional Laplacian, Critical nonlinearities, Convex–concave nonlinearities, Variational techniques, Mountain Pass Theorem

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