# American Institute of Mathematical Sciences

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

 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.

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:
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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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