Abstract:
We prove the existence of an unbounded branch of solutions to the nonlinear non-local equation (Equation presented) bifurcating from the first eigenvalue. Here (-Δ)sp denotes the fractional p-Laplacian and Ω ⊂ ℝ1 is a bounded regular domain. The proof of the bifurcation results relies in computing the Leray-Schauder degree by making an homotopy respect to s (the order of the fractional p-Laplacian) and then to use results of local case (that is s = 1) found in the paper of del Pino and Manasevich [J. Diff. Equ. 92(1991) (2), 226-251]. Finally, we give some application to an existence result. © European Mathematical Society.
Registro:
| Documento: |
Artículo
|
| Título: | Global bifurcation for fractional p-Laplacian and an application |
| Autor: | Del Pezzo, L.M.; Quaas, A. |
| Filiación: | CONICET, Departamento de Matematica, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria, Buenos Aires, 1428, Argentina Departamento de Matematica, Universidad Tecnica Federico Santa María Casilla V-110, Avda. España, Valparaíso, 1680, Chile
|
| Palabras clave: | Bifurcation; Existence results; Fractional p-Laplacian |
| Año: | 2016
|
| Volumen: | 35
|
| Número: | 4
|
| Página de inicio: | 411
|
| Página de fin: | 447
|
| DOI: |
http://dx.doi.org/10.4171/ZAA/1572 |
| Título revista: | Zeitschrift fur Analysis und ihre Anwendung
|
| Título revista abreviado: | Z. Anal. Anwend.
|
| ISSN: | 02322064
|
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02322064_v35_n4_p411_DelPezzo |
Referencias:
- Adams, R.A., Sobolev spaces (1975) Pure Appl. Math., 65. , New York: Academic Press
- Amghibech, S., On the discrete version of Picone's identity (2008) Discrete Appl. Math., 156 (1), pp. 1-10
- Anane, A., Simplicité et isolation de la première valeur propre du p-Laplacien avec poids (in French) (1987) C. R. Acad. Sci. Paris Sér. I Math., 305 (16), pp. 725-728
- Bandle, C., Reichel, W., Solutions of quasilinear second-order elliptic boundary value problems via degree theory (2004) Stationary Partial Differential Equations., 1, pp. 1-70. , eds.: M. Chipot et al.. Handb. Diff. Equ. Amsterdam: North-Holland
- Birindelli, I., Demengel, F., Bifurcation for singular fully nonlinear equations (2008) On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments, pp. 117-144. , eds.: A. Alvino et al.. Quad. Mat. 23. Caserta: Dept. Math., Seconda Univ. Napoli
- Bjorland, C., Caffarelli, L., Figalli, A., Non-local gradient dependent operators (2012) Adv. Math., 230 (4-6), pp. 1859-1894
- Bjorland, C., Caffarelli, L., Figalli, A., Nonlocal tug-of-war and the infinity fractional Laplacian (2012) Comm. Pure Appl. Math., 65 (3), pp. 337-380
- Bourgain, J., Brezis, H., Mironescu, P., Another look at Sobolev spaces (2000) Optimal Control and Partial Differential Equations, pp. 439-455. , Conference Paris, in honour of Professor Alain Bensoussan's 60th Birthday; eds.: J. L. Menaldi et al.. Amsterdam: IOS Press
- Brasco, L., Franzina, G., Convexity properties of Dirichlet integrals and Picone-type inequalities (2014) Kodai Math. J., 37 (3), pp. 769-799
- Brasco, L., Parini, E., The second eigenvalue of the fractional p-Laplacian Adv. Calc. Var., , to appear
- Busca, J., Esteban, M.J., Quaas, A., Nonlinear eigenvalues and bifurcation problems for Pucci's operators (2005) Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2), pp. 187-206
- Chambolle, A., Lindgren, E., Monneau, R., A Hölder infinity Laplacian (2012) ESAIM Control Optim. Calc. Var., 18 (3), pp. 799-835
- Chang, K., (2005) Methods in Nonlinear Analysis., , Springer Monogr. Math. Berlin: Springer
- Crandall, M., Rabinowitz, P., Bifurcation from simple eigenvalues (1971) J. Funct. Anal., 8, pp. 321-340
- Demengel, F., Demengel, G., Functional Spaces for the Theory of Elliptic Partial Differential Equations (2012) Transl. From the 2007 French Original by R. Erné, , Universitext. London: Springer
- Di Castro, A., Kuusi, T., Palatucci, G., Local behavior of fractional p-minimizers (2016) Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (5), pp. 1279-1299
- Di Nezza, E., Palatucci, G., Valdinoci, E., Hitchhiker's guide to the fractional Sobolev spaces (2012) Bull. Sci. Math., 136 (5), pp. 521-573
- Drábek, P., Solvability and bifurcations of nonlinear equations (1992) Pitman Res. Notes Math. Ser., 264. , Harlow: Longman
- Drábek, P., Girg, P., Takáč, P., Ulm, M., The Fredholm alternative for the p-Laplacian: Bifurcation from infinity, existence and multiplicity (2004) Indiana Univ. Math. J., 53 (2), pp. 433-482
- Fiscella, A., Molica Bisci, G., Servadei, R., Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems (2016) Bull. Sci. Math., 140, pp. 14-35
- Franzina, G., Palatucci, G., Fractional p-eigenvalues (2014) Riv. Mat. Univ. Parma (N. S.), 5 (2), pp. 373-386
- García-Melián, J., Sabina De Lis, J., A local bifurcation theorem for degenerate elliptic equations with radial symmetry (2002) J. Diff. Equ., 179 (1), pp. 27-43
- Grisvard, P., Elliptic problems in nonsmooth domains (1985) Monogr. Stud. Math., 24. , Boston MA: Pitman
- Iannizzotto, A., Mosconi, S., Squassina, M., Global Hölder regularity for the fractional p-Laplacian Rev. Mat. Iberoam., , to appear
- Ishii, H., Nakamura, G., A class of integral equations and approximation of p-Laplace equations (2010) Calc. Var. Partial Diff. Equ., 37 (3-4), pp. 485-522
- Kielhofer, H., Bifurcation theory. An introduction with applications to PDEs (2004) Appl. Math. Sci., 156. , New York: Springer
- Kuznetsov, Y., (2004) Elements of Applied Bifurcation Theory., 112. , Third edition. Appl. Math. Sci, New York: Springer
- Lindgren, E., Lindqvist, P., Fractional eigenvalues (2014) Calc. Var. Partial Diff. Equ., 49 (1-2), pp. 795-826
- Maz'ya, V., Shaposhnikova, T., On the bourgain, brezis, and mironescu theorem concerning limiting embeddings of fractional sobolev spaces (2002) J. Funct. Anal., 195 (2), pp. 230-238
- Molica Bisci, G., Servadei, R., A bifurcation result for non-local fractional equations (2015) Anal. Appl. (Singap.), 13 (4), pp. 371-394
- Papageorgiou, N., Kyritsi-Yiallourou, S., Handbook of applied analysis (2009) Adv. Mech. Math., 19. , New York: Springer
- Perera, K., Squassina, M., Yang, Y., Bifurcation and multiplicity results for critical fractional p-Laplacian problems (2015) Math. Nachr., 289, pp. 332-342
- Del Pino, M., Elgueta, M., Manasevich, R., A homotopic deformation along p of a Leray-Schauder degree result and existence for (u'|p-2u')' + f (t, u) = 0, u (0) = u (T) = 0, p > 1 (1989) J. Diff. Equ., 80 (1), pp. 1-13
- Del Pino, M., Manasevich, R., Global bifurcation from the eigenvalues of the p-Laplacian (1991) J. Diff. Equ., 92 (2), pp. 226-251
- Rabinowitz, P., Some global results for nonlinear eigenvalue problems (1971) J. Funct. Anal., 7, pp. 487-513
- Rabinowitz, P., Some aspects of nonlinear eigenvalue problems (1973) Rocky Mountain J. Math., 3, pp. 161-202
- Rabinowitz, P., (1975) Théorie du Degré Topologique et Applications à des Problèmes Aux Limites Non Linéaires (in French), , Paris: Universite Paris VI, Laboratoire Analyse Numerique
- Ros-Oton, X., Serra, J., The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2014) J. Math. Pures Appl., 101 (3-9), pp. 275-302
- Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator (2007) Comm. Pure Appl. Math., 60 (1), pp. 67-112
- Stein, E., Singular integrals and differentiability properties of functions (1970) Princeton Math. Ser., 30. , Princeton NJ: Princeton Univ. Press
- Struwe, M., (2008) Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems., 3 (34). , Fourth edition. Ergeb. Math. Grenzgeb, Berlin: Springer
Citas:
---------- APA ----------
Del Pezzo, L.M. & Quaas, A.
(2016)
. Global bifurcation for fractional p-Laplacian and an application. Zeitschrift fur Analysis und ihre Anwendung, 35(4), 411-447.
http://dx.doi.org/10.4171/ZAA/1572---------- CHICAGO ----------
Del Pezzo, L.M., Quaas, A.
"Global bifurcation for fractional p-Laplacian and an application"
. Zeitschrift fur Analysis und ihre Anwendung 35, no. 4
(2016) : 411-447.
http://dx.doi.org/10.4171/ZAA/1572---------- MLA ----------
Del Pezzo, L.M., Quaas, A.
"Global bifurcation for fractional p-Laplacian and an application"
. Zeitschrift fur Analysis und ihre Anwendung, vol. 35, no. 4, 2016, pp. 411-447.
http://dx.doi.org/10.4171/ZAA/1572---------- VANCOUVER ----------
Del Pezzo, L.M., Quaas, A. Global bifurcation for fractional p-Laplacian and an application. Z. Anal. Anwend. 2016;35(4):411-447.
http://dx.doi.org/10.4171/ZAA/1572