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doi: 10.3934/cpaa.2021177
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A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications

Dimitri Mugnai 1, , Kanishka Perera 2, and  Edoardo Proietti Lippi 3,,

1. 

Department of Ecology and Biology (DEB), Tuscia University, Largo dell'Università, 01100 Viterbo, Italy
2. 

Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901, USA
3. 

Department of Mathematics and Computer Science, University of Florence, Viale Morgagni 67/A, 50134 Firenze - Italy

* Corresponding author

Received  August 2021 Revised  October 2021 Early access October 2021

Fund Project: The first author is supported by the INdAM-GNAMPA Project 2020 "Equazioni alle derivate parziali: problemi e modelli" and by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017

We first prove that solutions of fractional p-Laplacian problems with nonlocal Neumann boundary conditions are bounded and then we apply such a result to study some resonant problems by means of variational tools and Morse theory.

Keywords: Fractional p-Laplacian, Neumann boundary conditions, cohomological local splitting.
Mathematics Subject Classification: Primary: 35R11, 58E05; Secondary: 35A15.
Citation: Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021177
References:
[1]

A. Audrito, J.-C. Felipe-Navarro and X. Ros-Oton, The Neumann problem for the fractional Laplacian: regularity up to the boundary, arXiv: 2006.10026. Google Scholar

[2]

B. Barrios, L. Montoro, I. Peral and F. Soria, Neumann conditions for the higher order $s$-fractional Laplacian $(-\Delta)^s u$ with $s > 1$, Nonlinear Anal., 193 (2020), 111-368.  doi: 10.1016/j.na.2018.10.012.  Google Scholar

[3]

K.-C. Chang and N. Ghoussoub, The Conley index and the critical groups via an extension of Gromoll-Meyer theory, Topol. Methods Nonlinear Anal., 7 (1996), 77-93.  doi: 10.12775/TMNA.1996.003.  Google Scholar

[4]

J.-N. Corvellec and Ha ntoute., Homotopical stability of isolated critical points of continuous functionals, Set-Valued Anal., 10 (2002), 143-164.  doi: 10.1023/A:1016544301594.  Google Scholar

[5]

M. Degiovanni, S. Lancelotti and K. Perera, Nontrivial solutions of $p-$superlinear $p-$Laplacian problems via a cohomological local splitting, Commun. Contemp. Math., 12 (2010), 475–486. doi: 10.1142/S0219199710003890.  Google Scholar

[6]

S. Dipierro, E. Proietti Lippi and E. Valdinoci, Linear theory for a mixed operator with Neumann conditions, arXiv: 2006.03850v1. Google Scholar

[7]

S. Dipierro, E. Proietti Lippi and E. Valdinoci, (Non)local logistic equations with Neumann conditions, arXiv: 2101.02315. Google Scholar

[8]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377–416. doi: 10.4171/RMI/942.  Google Scholar

[9]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139–174. doi: 10.1007/BF01390270.  Google Scholar

[10]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373–386.  Google Scholar

[11]

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p-$Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101–125. doi: 10.1515/acv-2014-0024.  Google Scholar

[12]

D. Motreanu, V. Motreanu and N, Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar

[13]

D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3,379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition. NoDEA. Nonlinear Differ. Equ. Appl. 19 (2012), 299–301. doi: 10.1007/s00030-011-0129-y.  Google Scholar

[14]

D. Mugnai and E. Proietti Lippi, Neumann fractional $p-$Laplacian: eigenvalues and existence results, Nonlinear Anal., 188 (2019), 455–474. doi: 10.1016/j.na.2019.06.015.  Google Scholar

[15]

D. Mugnai and E. Proietti Lippi, Linking over cones for the Neumann fractional $p-$Laplacian, J. Differ. Equ., 271 (2021), 797–820. doi: 10.1016/j.jde.2020.09.018.  Google Scholar

[16]

K. Perera, On the existence of ground state solutions to critical growth problems nonresonant at zero, arXiv: 2106.12170. Google Scholar

[17]

K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p-$Laplacian Type Operators, Math. Surveys Monogr. 161, 2010. doi: 10.1090/surv/161.  Google Scholar

[18]

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

[19]

Z. Vondra$\check{c}$ek, A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem, arXiv: 1909.10687.  Google Scholar

show all references

References:
[1]

A. Audrito, J.-C. Felipe-Navarro and X. Ros-Oton, The Neumann problem for the fractional Laplacian: regularity up to the boundary, arXiv: 2006.10026. Google Scholar

[2]

B. Barrios, L. Montoro, I. Peral and F. Soria, Neumann conditions for the higher order $s$-fractional Laplacian $(-\Delta)^s u$ with $s > 1$, Nonlinear Anal., 193 (2020), 111-368.  doi: 10.1016/j.na.2018.10.012.  Google Scholar

[3]

K.-C. Chang and N. Ghoussoub, The Conley index and the critical groups via an extension of Gromoll-Meyer theory, Topol. Methods Nonlinear Anal., 7 (1996), 77-93.  doi: 10.12775/TMNA.1996.003.  Google Scholar

[4]

J.-N. Corvellec and Ha ntoute., Homotopical stability of isolated critical points of continuous functionals, Set-Valued Anal., 10 (2002), 143-164.  doi: 10.1023/A:1016544301594.  Google Scholar

[5]

M. Degiovanni, S. Lancelotti and K. Perera, Nontrivial solutions of $p-$superlinear $p-$Laplacian problems via a cohomological local splitting, Commun. Contemp. Math., 12 (2010), 475–486. doi: 10.1142/S0219199710003890.  Google Scholar

[6]

S. Dipierro, E. Proietti Lippi and E. Valdinoci, Linear theory for a mixed operator with Neumann conditions, arXiv: 2006.03850v1. Google Scholar

[7]

S. Dipierro, E. Proietti Lippi and E. Valdinoci, (Non)local logistic equations with Neumann conditions, arXiv: 2101.02315. Google Scholar

[8]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377–416. doi: 10.4171/RMI/942.  Google Scholar

[9]

E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139–174. doi: 10.1007/BF01390270.  Google Scholar

[10]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373–386.  Google Scholar

[11]

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p-$Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101–125. doi: 10.1515/acv-2014-0024.  Google Scholar

[12]

D. Motreanu, V. Motreanu and N, Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar

[13]

D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3,379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition. NoDEA. Nonlinear Differ. Equ. Appl. 19 (2012), 299–301. doi: 10.1007/s00030-011-0129-y.  Google Scholar

[14]

D. Mugnai and E. Proietti Lippi, Neumann fractional $p-$Laplacian: eigenvalues and existence results, Nonlinear Anal., 188 (2019), 455–474. doi: 10.1016/j.na.2019.06.015.  Google Scholar

[15]

D. Mugnai and E. Proietti Lippi, Linking over cones for the Neumann fractional $p-$Laplacian, J. Differ. Equ., 271 (2021), 797–820. doi: 10.1016/j.jde.2020.09.018.  Google Scholar

[16]

K. Perera, On the existence of ground state solutions to critical growth problems nonresonant at zero, arXiv: 2106.12170. Google Scholar

[17]

K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p-$Laplacian Type Operators, Math. Surveys Monogr. 161, 2010. doi: 10.1090/surv/161.  Google Scholar

[18]

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

[19]

Z. Vondra$\check{c}$ek, A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem, arXiv: 1909.10687.  Google Scholar

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