Article Contents
Article Contents

# A nonlocal concave-convex problem with nonlocal mixed boundary data

• * Corresponding author
The first author is supported by research grants MTM2013-40846-P and MTM2016-80474-P, MINECO, Spain
• The aim of this paper is to study the following problem

$(P_{\lambda}) \equiv\left\{\begin{array}{rcll}(-\Delta)^s u& = &\lambda u^{q}+u^{p}&{\text{ in }}\Omega,\\ u&>&0 &{\text{ in }} \Omega, \\ \mathcal{B}_{s}u& = &0 &{\text{ in }} \mathbb{R}^{N}\backslash \Omega,\end{array}\right.$

with $0<q<1<p$, $N>2s$, $λ> 0$, $Ω \subset \mathbb{R}^{N}$ is a smooth bounded domain,

$(-Δ)^su(x) = a_{N,s}\;P.V.∈t_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,$

$a_{N,s}$ is a normalizing constant, and $\mathcal{B}_{s}u = uχ_{Σ_{1}}+\mathcal{N}_{s}uχ_{Σ_{2}}.$ Here, $Σ_{1}$ and $Σ_{2}$ are open sets in $\mathbb{R}^{N}\backslash Ω$ such that $Σ_{1} \cap Σ_{2} = \emptyset$ and $\overline{Σ}_{1}\cup \overline{Σ}_{2} = \mathbb{R}^{N}\backslash Ω.$

In this setting, $\mathcal{N}_{s}u$ can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and $\mathcal{B}_{s}u$ is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem ($P_{λ}$) for suitable ranges of $λ$ and $p$ and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

Mathematics Subject Classification: Primary: 35R11, 35A15, 35A16; Secondary: 35J61, 60G22.

 Citation:

•  S. Alama , Semilinear elliptic equation with sublinear indefinite nonlinearities, Adv. Differential Equation, 4 (1999) , 813-842. A. Ambrosetti , Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992) , 1-139. A. Ambrosetti , H. Brezis  and  G. Cerami , Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994) , 519-543. A. Ambrosetti  and  P.H. Rabinowitz , Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973) , 349-381. D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009. J. G. Azorero  and  I. Peral , Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term, Trans. Am. Math. Soc, 323 (1991) , 877-895. B. Barrios , E. Colorado , R. Servadei  and  F. Soria , A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015) , 875-900. B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505. B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math, 16 (2014), 1350046, 29 pp. H. Brezis  and  S. Kamin , Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992) , 87-106. C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^{N}$ with critical growth and convex nonlinearities, arXiv: 1609.01911. C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016. L. Caffarelli  and  L. Silvestre , An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007) , 1245-1260. L. Caffarelli  and  L. Silvestre , Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011) , 59-88. E. Colorado  and  I. Peral , Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003) , 468-507. M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015. E. Di Nezza , G. Palatucci  and  E. Valdinoci , Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012) , 521-573. S. Dipierro , M. Medina , I. Peral  and  E. Valdinoci , Bifurcation results for a fractional elliptic equation with critica exponent in $\mathbb{R}^N$, Manuscripta Math., 153 (2017) , no.1-230. S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^{N}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 2017. S. Dipierro , X. Ros-Oton  and  E. Valdinoci , Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017) , 377-416. N. Ghoussoub  and  D. Preiss , A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 6 (1989) , 321-330. M. Grossi  and  F. Pacella , Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A., 116 (1990) , 23-43. N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag. T. Leonori , I. Peral , A. Primo  and  F. Soria , Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015) , 6031-6068. A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016. X. Ros-Oton , Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016) , 3-26. X. Ros-Oton  and  J. Serra , The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014) , 275-302. R. Servadei  and  E. Valdinoci , Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012) , 887-898. R. Servadei  and  E. Valdinoci , Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013) , 2105-2137. R. Servadei  and  E. Valdinoci , Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014) , 133-154. G. Stampacchia , Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965) , 189-258. M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin Heidelberg, 1990.