A critical fractional equation with concave-convex power nonlinearities

By Begoña Barrios, Eduardo Colorado, Raffaella Servadei, Fernando Soria, F. Soria. 2013

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In this work we study the following fractional critical problem $$ (P_{\lambda})=\left\{\begin{array}{ll} (-\Delta)^s u=\lambda u^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} \Omega\\ u=0 & \mbox{in} \RR^n\setminus \Omega\,, \end{array}\right. $$ where $\Omega\subset \mathbb{R}^n$ is a regular bounded domain, $\lambda>0$, $0 2s$. Here $(-\Delta)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-\Delta)^s u(x)={\rm P. V.} \int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n. $$ Our main results show the existence and multiplicity of solutions to problem $(P_\lambda)$ for different values of $\lambda$. The dependency on this parameter changes according to whether we consider the concave power case ($0

Subfields: Applied Mathematics, Modeling and Simulation

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Citations in the corpus

On Dirichlet problem for fractional p-Laplacian with singular non-linearity (2016, background): “The existence and multiplicity results for non-local operators like fractional Laplacian with combination of convex and concave type non linearity like uq + λup, p, q > 0 is studied in [4, 6, 36, 37, 44, 45].”
Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “respectively in [4, 20] for a positive weak solution and [30, 31] for a weak solution which could change sign.”
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “Last but not least, fractional elliptic problems with critical growth, in a bounded domain, have been studied by some authors in the last years, see [6, 7, 18, 28, 39, 41] and references therein.”
Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, unknown): “Our argument is similar to that developed in Proposition 3.2 of [10] but we give the details for the reader’s convenience.”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (2015, background): “The literature on fractional operators and their applications is very huge, here we just mention a few, see for example [10–20], especially [21] and [22,23] for two different fractional Laplacian involving concave–convex nonlinearities.”
The Nehari manifold for fractional systems involving critical nonlinearities (2015, background): “We refer to [12, 23, 26, 27, 29, 32] for the subcritical case and to [3, 4, 11, 14, 15, 22, 24, 28] for the critical case.”
A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝN (2016, background): “In [31], the authors established the principle of concentration compactness in fractional Sobolev spaces Ds(RN) by using profile decomposition; see also [9] for its applications.”
A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, unknown): “4] (see also the nonlocal counterpart in [6]).”
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity (2014, background): “. Concerning the critical case, in [5] the eﬀects of lower order perturbations are studied for the existence of positive solutions of critical elliptic problems involving the fractional Laplacian. In [6], see also the references therein, existence and multiplicity results are established for a criti”
On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents (2016, background): “For example, [7, 16, 17, 18, 19, 29, 31, 48] studied the critical problems on bounded or unbounded domains, [24, 32] investigated the quasilinear critical growth problems driven by p-Laplacian, [23] studied the regularity of stable solutions of elliptic problems on Riemannian manifold, [22, 25, 27, ”
Critical Nonlocal Systems with Concave-Convex Powers (2016, background): “On the other hand, the fractional problems for p = 2 have been investigated by many researchers, see, for example, [2, 6, 23] for the critical case and [11] for the fractional Kirchhoff type problem.”