Artículo

Saintier, N.; Silva, A."Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN" (2017) Nonlinear Differential Equations and Applications. 24(2)
Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

The purpose of this paper is to formulate sufficient existence conditions for a critical equation involving the p(x)-Laplacian of the form (0.1) below posed in RN. This equation is critical in the sense that the source term has the form K(x) | u| q ( x ) - 2u with an exponent q that can be equal to the critical exponent p∗ at some points of RN including at infinity. The sufficient existence condition we find are local in the sense that they depend only on the behaviour of the exponents p and q near these points. We stress that we do not assume any symmetry or periodicity of the coefficients of the equation and that K is not required to vanish in some sense at infinity like in most existing results. The proof of these local existence conditions is based on a notion of localized best Sobolev constant at infinity and a refined concentration-compactness at infinity. © 2017, Springer International Publishing.

Registro:

Documento: Artículo
Título:Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN
Autor:Saintier, N.; Silva, A.
Filiación:Departamento de Matemática, FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428), Buenos Aires, Argentina
Instituto de Matemática Aplicada San Luis, IMASL, Universidad Nacional de San Luis and CONICET, Ejercito de los Andes 950, San Luis, D5700HHW, Argentina
Palabras clave:Concentration compactness; Critical exponents; Sobolev embedding; Variable exponents
Año:2017
Volumen:24
Número:2
DOI: http://dx.doi.org/10.1007/s00030-017-0441-2
Handle:http://hdl.handle.net/20.500.12110/paper_10219722_v24_n2_p_Saintier
Título revista:Nonlinear Differential Equations and Applications
Título revista abreviado:Nonlinear Diff. Equ. Appl.
ISSN:10219722
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10219722_v24_n2_p_Saintier

Referencias:

  • Multi-bump solutions for a class of quailinear problems involving variable exponents (to appear), , Alves, C.O., Ferreira, M.C
  • (2006) Existence of solutions for a class of problems in Rn involving the p(x) -Laplacian. In: Contributions to Nonlinear Analysis, Volume 66 of Programming Nonlinear Differential Equations Applications, , Alves, C.O., Souto, M.A.S
  • Alves, C.O., Existence of solution for a degenerate p(x)-Laplacian equation in Rn (2008) J. Math. Anal. Appl., 345, pp. 731-742
  • Alves, C.O., Existence of radial solutions for a class of p(x) -laplacian with critical growth (2010) Differ. Integral Equ., 23 (1-2), pp. 113-123
  • Alves, O., Ferreira, M.C., Nonlinear perturbations of a p(x) -Laplacian equation with critical growth in Rn (2014) Math. Nachr., 287 (8-9), pp. 849-868
  • Alves, C.O., Ferreira, M.C., Existence of solutions for a class of p(x) -Laplacian equations involving a concave-convex nonlinearity with critical growth in Rn (2015) Topol. Methods Nonlinear Anal., 45 (2), pp. 399-422
  • Aubin, T., Problèmes isopérimétriques et espaces de Sobolev (1976) J. Differ. Geom., 11 (4), pp. 573-598
  • Critical Sobolev embeddings in variable exponent spaces and applications, , Bonder, J.F., Saintier, N., Silva, A
  • Bonder, J.F., Saintier, N., Silva, A., Existence of solution to a critical equation with variable exponent (2012) Ann. Acad. Sci. Fenn. Math., 37, pp. 579-594
  • Bonder, J.F., Saintier, N., Silva, A., On the Sobolev embedding theorem for variable exponent spaces in the critical range (2012) J. Differ. Equ., 253 (5), pp. 1604-1620
  • Bonder, J.F., Saintier, N., Silva, A., On the Sobolev trace theorem for variable exponent spaces in the critical range (2014) Annali di Matematica Pura ed Aplicata, 193 (6), pp. 1607-1628
  • Bonder, J.F., Saintier, N., Silva, A., Existence of solution to a critical trace equation with variable exponent (2015) Asymp. Anal., 93 (1-2), pp. 161-185
  • Bonder, J.F., Silva, A., Concentration-compactness principle for variable exponent spaces and applications (2010) Electron. J. Differ. Equ., 141, p. 18
  • Brézis, H., Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents (1983) Commun. Pure Appl. Math., 36 (4), pp. 437-477
  • Chabrowski, J., Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents (1995) Calc. Var. Partial Differ. Equ., 3 (4), pp. 493-512
  • Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration (2006) SIAM J. Appl. Math., 66 (4), pp. 1383-1406. , (electronic)
  • Diening, L., Harjulehto, P., Hästö, P., Ružička, M., (2011) Lebesgue and Sobolev spaces with Variable Exponents, Volume 2017 of Lecture Notes in Mathematics, , Springer, Heidelberg
  • Druet, O., Hebey, E., Robert, F., (2004) Blow-Up Theory for Elliptic PDEs in Riemannian Geometry, Volume 45 of Mathematical Notes, , Princeton University Press, Princeton
  • Fan, X., A constrained minimization problem involving the p(x) -Laplacian in Rn (2008) Nonlinear Anal., 69, pp. 3661-3670
  • Fan, X., p(x) -Laplacian equations in Rn with periodic data and nonperiodic perturbations (2008) J. Math. Anal. Appl., 341, pp. 103-119
  • Fan, X., Xiaoyou, H., Existence and multiplicity of solutions for p(x)-Laplacian equations in Rn (2004) Nonlinear Anal., 59, pp. 173-188
  • Futamura, T., Mizuta, Y., Shimomura, T., Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent (2010) J. Math. Anal. Appl., 366, pp. 391-417
  • Gilbarg, D., Trudinger, N.S., (2001) Ellitpic Partial Differential Equations of Second-Order, Volume 1748 of Classics in Mathematics, , Springer, Berlin
  • Harjulehto, P., Hästö, P., Le, U.V., Matti, N., Overview of differential equations with non-standard growth (2010) Nonlinear Anal., 72, pp. 4551-4574
  • Hästö, P.A., Local-to-global results in variable exponent spaces (2009) Math. Res. Lett., 16 (2), pp. 263-278
  • Lee, J.M., (1997) Riemannian Manifolds, An Introduction to Curvature, Volume of 176 Graduate Texts in Mathematics, , Springer, Berlin
  • Liang, S., Zhang, J., Multiple solutions for noncooperative p(x) -Laplacian equations in Rn involving the critical exponent (2013) J. Math. Anal. Appl., 403, pp. 344-356
  • Lions, P.-L., Symétrie et compacité dans les espaces de sobolev (1982) J. Funct. Anal., 49, pp. 315-334
  • Lions, P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I (1985) Rev. Mat. Iberoam., 1 (1), pp. 145-201
  • Mizuta, Y., Ohno, T., Shimomura, T., Shioji, N., Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x) -Laplacian and its critical exponent (2010) Ann. Acad. Sci. Fenn. Math., 35 (1), pp. 115-130
  • Nekvinda, A., Equivalence of l pn norms and shift operators (2002) Math. Inequal. Appl., 5 (4), pp. 711-723
  • Radulescu, V.D., Nonlinear elliptic equations with variable eponent: old and new (2015) Nonlinear Anal., 121, pp. 336-369
  • Radulescu, V.D., Repovš, D.D., (2015) Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics, , CRC Press, Boca Raton
  • Ružička, M., (2000) Electrorheological Fluids: Modeling and Mathematical Theory, Volume 1748 of Lecture Notes in Mathematics, , Springer, Berlin
  • Saintier, N., Asymptotic estimates and blow-up theory for critical equations involving the p -Laplacian (2006) Calc. Var. Partial Differ. Equ., 25 (3), pp. 299-331
  • Saintier, N., Estimates of the best Sobolev constant of the embedding of bv(ω) into l1(∂ω) and related shape optimization problems (2008) Nonlinear Anal. TMA, 69, pp. 2479-2491
  • Saintier, N., Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds (2009) Calc. Var. Partial Differ. Equ., 35, pp. 385-407
  • Strauss, W.A., Existence of solitary waves in higher dimensions (1977) Commun. Math. Phys., 55, pp. 149-162
  • Yongqiang, F., The principle of concentration compactness in Lp ( x ) spaces and its application (2009) Nonlinear Anal., 71 (5-6), pp. 1876-1892
  • Yongqiang, F., Shan, Y., On the removability of isolated singular points for elliptic equations involving variable exponent (2016) Adv. Nonlinear Anal., 5 (2), pp. 1-12
  • Yongqiang, F., Zhang, X., A multiplicity result for p(x)-Laplacian problem in Rn (2009) Nonlinear Anal., 70, pp. 2261-2269
  • Yongqiang, F., Zhang, X., Multiple solutions for a class of p(x) -Laplacian equations in involving the critical exponent. Proc. R. Soc. Lond. Ser. A Math. Phys (2010) Eng. Sci., 466 (2118), pp. 1667-1686
  • Yongqiang, F., Zhang, X., Solutions of p(x) -Laplacian equations with critical exponent and perturbations in Rn (2012) Electron. J. Differ. Equ., 2012 (120), pp. 1-14

Citas:

---------- APA ----------
Saintier, N. & Silva, A. (2017) . Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN. Nonlinear Differential Equations and Applications, 24(2).
http://dx.doi.org/10.1007/s00030-017-0441-2
---------- CHICAGO ----------
Saintier, N., Silva, A. "Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN" . Nonlinear Differential Equations and Applications 24, no. 2 (2017).
http://dx.doi.org/10.1007/s00030-017-0441-2
---------- MLA ----------
Saintier, N., Silva, A. "Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN" . Nonlinear Differential Equations and Applications, vol. 24, no. 2, 2017.
http://dx.doi.org/10.1007/s00030-017-0441-2
---------- VANCOUVER ----------
Saintier, N., Silva, A. Local existence conditions for an equations involving the p(x) -Laplacian with critical exponent in RN. Nonlinear Diff. Equ. Appl. 2017;24(2).
http://dx.doi.org/10.1007/s00030-017-0441-2