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

A nonlocal concave-convex problem with nonlocal mixed boundary data

Boumediene Abdellaoui 1, , Abdelrazek Dieb 2, and  Enrico Valdinoci 3,4,5,6,,

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
$0<q<1<p$
,
$N>2s$
,
$λ> 0$
,
$Ω \subset \mathbb{R}^{N}$
 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.
Keywords: Integro differential operators, fractional Laplacian, weak solutions, mixed boundary condition, multiplicity of positive solution.
Mathematics Subject Classification: Primary: 35R11, 35A15, 35A16; Secondary: 35J61, 60G22.
Citation: Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure and 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. 

[2]

A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992), 1-139. 

[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. 

[4]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. 

[5]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009.

[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. 

[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. 

[8]

B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505.

[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.

[10]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. 

[11]

C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^{N}$ with critical growth and convex nonlinearities, arXiv: 1609.01911.

[12]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016.

[13]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. 

[14]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. 

[15]

E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003), 468-507. 

[16]

M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015.

[17]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[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. 

[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.

[20]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416. 

[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. 

[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. 

[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.

[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. 

[25]

A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016.

[26]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. 

[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. 

[28]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898. 

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. 

[30]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. 

[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. 

[32]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin Heidelberg, 1990.

show all references

References:
[1]

S. Alama, Semilinear elliptic equation with sublinear indefinite nonlinearities, Adv. Differential Equation, 4 (1999), 813-842. 

[2]

A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.), 49 (1992), 1-139. 

[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. 

[4]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. 

[5]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd, edition, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, 2009.

[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. 

[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. 

[8]

B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, arXiv: 1607.01505.

[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.

[10]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. 

[11]

C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^{N}$ with critical growth and convex nonlinearities, arXiv: 1609.01911.

[12]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer; Unione Matematica Italiana, Bologna, 2016.

[13]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. 

[14]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. 

[15]

E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal, 199 (2003), 468-507. 

[16]

M. Cozzi, Qualitative Properties of Solutions of Nonlinear Anisotropic PDEs in Local and Nonlocal Settings, PhD thesis, 2015.

[17]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[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. 

[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.

[20]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam, 33 (2017), 377-416. 

[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. 

[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. 

[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.

[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. 

[25]

A. C. Ponce, Elliptic PDEs, Measures and Capacities, Tracts in Mathematics 23, European Mathematical Society (EMS), Zurich, 2016.

[26]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. 

[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. 

[28]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl, 389 (2012), 887-898. 

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. 

[30]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. 

[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. 

[32]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin Heidelberg, 1990.

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