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Abstract:

In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the p(x)-Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration - compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem. © 2015 - IOS Press and the authors. All rights reserved.

Registro:

Documento: Artículo
Título:Existence of solution to a critical trace equation with variable exponent
Autor:Bonder, J.F.; Saintier, N.; Silva, A.
Filiación:IMAS, CONICET, Universidad de Buenos Aires, Pabellón I (1428), Buenos Aires, Argentina
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina
Instituto de Ciencias, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina
Palabras clave:concentration compactness; critical exponents; Sobolev embedding; variable exponents; Asymptotic analysis; Concentration-compactness principle; Critical exponent; Existence of Solutions; Mountain pass theorem; Nontrivial solution; Sobolev embedding; Variable exponent Sobolev space; Variable exponents; Sobolev spaces
Año:2015
Volumen:93
Número:1-2
Página de inicio:161
Página de fin:185
DOI: http://dx.doi.org/10.3233/ASY-151289
Título revista:Asymptotic Analysis
Título revista abreviado:Asymptotic Anal
ISSN:09217134
CODEN:ASANE
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09217134_v93_n1-2_p161_Bonder

Referencias:

  • Adimurthi, Yadava, S.L., Positive solution for Neumann problem with critical nonlinearity on boundary (1991) Comm. Partial Differential Equations, 16 (11), pp. 1733-1760
  • Aubin, T., Problèmes isopérimétriques et espaces de Sobolev (1975) C. R. Acad. Sci. Paris Sér. A-B, 280 (5), pp. A279-A281. , Aii
  • Aubin, T., Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire (1976) J. Math. Pures Appl. (9), 55 (3), pp. 269-296
  • Aubin, T., (1998) Some Nonlinear Problems in Riemannian Geometry, , Springer Monographs in Mathematics, Springer-Verlag, Berlin
  • Brézis, H., Lieb, E., A relation between pointwise convergence of functions and convergence of functionals (1983) Proc. Amer. Math. Soc., 88 (3), pp. 486-490
  • Brézis, H., Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents (1983) Comm. Pure Appl. Math., 36 (4), pp. 437-477
  • Demengel, F., Hebey, E., On some nonlinear equations involving the p-Laplacian with critical Sobolev growth (1998) Adv. Differential Equations, 3 (4), pp. 533-574
  • Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M., (2011) Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. , Springer, Heidelberg
  • Djadli, Z., Hebey, E., Ledoux, M., Paneitz-type operators and applications (2000) Duke Math. J., 104 (1), pp. 129-169
  • Druet, O., Generalized scalar curvature type equations on compact Riemannian manifolds (2000) Proc. Roy. Soc. Edinburgh Sect. A, 130 (4), pp. 767-788
  • Druet, O., Hebey, E., The AB program in geometric analysis: Sharp Sobolev inequalities and related problems (2002) Mem. Amer. Math. Soc., 160 (761), pp. viii-98
  • Escobar, J.F., Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary (1992) Ann. Math. (2), 136 (1), pp. 1-50
  • Esposito, P., Robert, F., Mountain pass critical points for Paneitz-Branson operators (2002) Calc. Var. Partial Differential Equations, 15 (4), pp. 493-517
  • Evans, L.C., (1990) Weak Convergence Methods for Nonlinear Partial Differential Equations, In: CBMS Regional Conference Series in Mathematics, 74. , Published for the Conference Board of the Mathematical Sciences, Washington, DC
  • Faget, Z., Best constants in Sobolev inequalities on Riemannian manifolds in the presence of symmetries (2002) Potential Anal., 17 (2), pp. 105-124
  • Fan, X., Boundary trace embedding theorems for variable exponent Sobolev spaces (2008) J. Math. Anal. Appl., 339 (2), pp. 1395-1412
  • Fan, X., Zhao, D., On the spaces Lp(x)(?) and Wm,p(x)(?) (2001) J. Math. Anal. Appl., 263 (2), pp. 424-446
  • Fernández Bonder, J., Saintier, N., Estimates for the Sobolev trace constant with critical exponent and applications (2008) Ann. Mat. Pura Appl. (4), 187 (4), pp. 683-704
  • Fernández Bonder, J., Saintier, N., Silva, A., Existence of solution to a critical equation with variable exponent (2012) Ann. Acad. Sci. Fenn. Math., 37, pp. 579-594
  • Fernández Bonder, J., Saintier, N., Silva, A., On the Sobolev embedding theorem for variable exponent spaces in the critical range (2012) J. Differential Equations, 253 (5), pp. 1604-1620
  • Fernández Bonder, J., Saintier, N., Silva, A., On the Sobolev trace theorem for variable exponent spaces in the critical range (2014) Ann. Mat. Pura Appl. (4), 193 (6), pp. 1607-1628
  • Fernández Bonder, J., Silva, A., Concentration-compactness Principle for Variable Exponent Spaces and Applications, p. 18. , Electron. J. Differential Equations (2010), Paper No. 141
  • Fu, Y., The principle of concentration compactness in Lp(x) spaces and its application (2009) Nonlinear Anal., 71 (5-6), pp. 1876-1892
  • Gray, A., (1990) Tubes, , Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA
  • Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S., The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values (2006) Potential Anal., 25 (3), pp. 205-222
  • Hebey, E., Vaugon, M., Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth (1994) J. Funct. Anal., 119 (2), pp. 298-318
  • Kovácik, O., Rákosník, J., On spaces Lp(x) and Wk,p(x) (1991) Czechoslovak Math. J., 41 (116), pp. 592-618
  • 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
  • Nazaret, B., Best constant in Sobolev trace inequalities on the half-space (2006) Nonlinear Anal., 65 (10), pp. 1977-1985
  • Saintier, N., Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian (2006) Calc. Var. Partial Differential Equations, 25 (3), pp. 299-331
  • Saintier, N., Estimates of the best Sobolev constant of the embedding of BV (?) into L1() and related shape optimization problems (2008) Nonlinear Anal., 69 (8), pp. 2479-2491
  • Saintier, N., Best constant in critical Sobolev inequalities of second-order in the presence of symmetries (2010) Nonlinear Anal., 72 (2), pp. 689-703
  • Schoen, R., Conformal deformation of a Riemannian metric to constant scalar curvature (1984) J. Differential Geom., 20 (2), pp. 479-495

Citas:

---------- APA ----------
Bonder, J.F., Saintier, N. & Silva, A. (2015) . Existence of solution to a critical trace equation with variable exponent. Asymptotic Analysis, 93(1-2), 161-185.
http://dx.doi.org/10.3233/ASY-151289
---------- CHICAGO ----------
Bonder, J.F., Saintier, N., Silva, A. "Existence of solution to a critical trace equation with variable exponent" . Asymptotic Analysis 93, no. 1-2 (2015) : 161-185.
http://dx.doi.org/10.3233/ASY-151289
---------- MLA ----------
Bonder, J.F., Saintier, N., Silva, A. "Existence of solution to a critical trace equation with variable exponent" . Asymptotic Analysis, vol. 93, no. 1-2, 2015, pp. 161-185.
http://dx.doi.org/10.3233/ASY-151289
---------- VANCOUVER ----------
Bonder, J.F., Saintier, N., Silva, A. Existence of solution to a critical trace equation with variable exponent. Asymptotic Anal. 2015;93(1-2):161-185.
http://dx.doi.org/10.3233/ASY-151289