American Institute of Mathematical Sciences

May  2018, 17(3): 1103-1120. doi: 10.3934/cpaa.2018053

A nonlocal concave-convex problem with nonlocal mixed boundary data

 1 Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria 2 Département de Mathématiques, Université Ibn Khaldoun, Tiaret, Tiaret 14000, Algeria 3 University of Melbourne, School of Mathematics and Statistics, Peter Hall Building, Parkville, Melbourne VIC 3010, Australia 4 School of Mathematics and Statistics, 35 Stirling Highway, Crawley, Perth WA 6009, Australia 5 Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50,20133 Milan, Italy 6 Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1,27100 Pavia, Italy

* Corresponding author

Received  November 2016 Revised  October 2017 Published  January 2018

Fund Project: 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 , $N>2s$, $λ> 0$, $Ω \subset \mathbb{R}^{N}$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. Citation: Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053 References:  [1] S. Alama, Semilinear elliptic equation with sublinear indefinite nonlinearities, Adv. Differential Equation, 4 (1999), 813-842. Google Scholar [2] A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992), 1-139. Google Scholar [3] 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. Google Scholar [4] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar [5] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505. Google Scholar [9] 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. Google Scholar [10] H. Brezis and S. Kamin, Sublinear elliptic equations in$\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. Google Scholar [11] C. Bucur and M. Medina, A fractional elliptic problem in$\mathbb{R}^{N}$with critical growth and convex nonlinearities, arXiv: 1609.01911. Google Scholar [12] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016. Google Scholar [13] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. Google Scholar [14] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. Google Scholar [15] E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003), 468-507. Google Scholar [16] M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015. Google Scholar [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416. Google Scholar [21] 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. Google Scholar [22] 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. Google Scholar [23] 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. Google Scholar [24] 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. Google Scholar [25] A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016. Google Scholar [26] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. Google Scholar [27] 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. Google Scholar [28] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898. Google Scholar [29] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar [30] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. Google Scholar [31] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. Google Scholar [32] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin Heidelberg, 1990. Google Scholar show all references References:  [1] S. Alama, Semilinear elliptic equation with sublinear indefinite nonlinearities, Adv. Differential Equation, 4 (1999), 813-842. Google Scholar [2] A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992), 1-139. Google Scholar [3] 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. Google Scholar [4] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar [5] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505. Google Scholar [9] 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. Google Scholar [10] H. Brezis and S. Kamin, Sublinear elliptic equations in$\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. Google Scholar [11] C. Bucur and M. Medina, A fractional elliptic problem in$\mathbb{R}^{N}$with critical growth and convex nonlinearities, arXiv: 1609.01911. Google Scholar [12] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016. Google Scholar [13] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. Google Scholar [14] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. Google Scholar [15] E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003), 468-507. Google Scholar [16] M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015. Google Scholar [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416. Google Scholar [21] 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. Google Scholar [22] 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. Google Scholar [23] 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. Google Scholar [24] 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. Google Scholar [25] A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016. Google Scholar [26] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. Google Scholar [27] 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. Google Scholar [28] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898. Google Scholar [29] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar [30] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. Google Scholar [31] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. Google Scholar [32] M. 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