Carole Louis-Rose^[1] and Jean Vélin^[2]

On a non-existence result involving the fractional \(p\)-Laplacian

Pages: 345-355
Received: 8 September 2015   
Accepted in revised form: 12 January 2016
Mathematics Subject Classification (2010): 35R11.
Keywords: Non-existence, weak solution, fractional \(p\)-Laplacian.
Author address:
[1],[2] : Department of Mathematics and Computer, Laboratory CEREGMIA, University of Antilles, Campus of Fouillole, Pointe-à-Pitre, 97159, Guadeloupe (FWI)

Full Text (PDF)

Abstract: We consider a nonlocal problem involving the fractional \(p\)-Laplacian operator in bounded smooth domains. A non-existence result is obtained via a comparison process. This result extends those done for the fractional Laplacian.

References

[1] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré (C) Non Linear Analysis 31 (2014), 23-53. MR3165278
[2] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations Abel Symp., 7, Springer, Heidelberg 2012, 37-52. MR3289358
[3] M. D'Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl. 66 (2013), 1245-1260. MR3096457
[4] F. De Thélin, Quelques résultats d'existence et de non-existence pour une EDP elliptique non linéaire, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 911-914. MR0774666
[5] F. De Thélin, Résultats d'existence et de non-existence pour la solution positive et bornée d'une e.d.p. elliptique non linéaire, Ann. Fac. Sci. Toulouse Math. 8 (1986/87), 375-389. MR0948761
[6] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional \(p\)-minimizers, Ann. Inst. H. Poincaré (C) Non Linear Analysis (2015), in press, doi:10.1016/j.anihpc.2015.04.003.
[7] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. MR2944369
[8] R. Ferreira and M. Pérez-Llanos, Limit problems for a fractional \(p\)-Laplacian as \(p\to\infty\), NoDEA Nonlinear Differential Equations Appl., to appear.
[9] G. Franzina and G. Palatucci, Fractional \(p\)-eigenvalues, Riv. Mat. Univ. Parma 5 (2014), 373-386. MR3307955
[10] A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional \(p\)-Laplacian problems via Morse theory, Adv. Calc. Var. (2014), to appear, DOI 10.1515/acv-2014-0024.
[11] R. Ignat, Pohozaev-type identities for an elliptic equation, in "Singularities in PDE and the calculus of variations", CRM Proc. Lecture Notes, 44, Stanley Alama, Lia Bronsard and Peter J . Sternberg (eds), Amer. Math. Soc., Providence, RI, 2008, 75-88. MR2528735
[12] R. Lehrer, L. A. Maia and M. Squassina, On fractional \(p\)-Laplacian problems with weight, Differential Integral Equations 28 (2015), 15-28. MR3299115
[13] M. Otani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988), 140-159. MR0923049
[14] S. I. Pohozaev, On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\) (Russian), Dokl. Akad. Nauk SSSR 165 (1965), 36-39. MR0192184
[15] X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris 350 (2012), 505-508. MR2929057
[16] X. Ros-Oton and J. Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst. 35 (2015), 2131-2150. MR3294242
[17] X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations 40 (2015), 115-133. MR3268923
[18] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), 587-628. MR3211861