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Abstract:

We study the following elliptic problem - A(u) = λuq with Dirichlet boundary conditions, where A(u)(x)=Δu(x)χD1(x)+Δpu(x)χD2(x) is the Laplacian in one part of the domain, D1, and the p-Laplacian (with p> 2) in the rest of the domain, D2. We show that this problem exhibits a concave–convex nature for 1 < q< p- 1. In fact, we prove that there exists a positive value λ∗ such that the problem has no positive solution for λ> λ∗ and a minimal positive solution for 0 < λ< λ∗. If in addition we assume that p is subcritical, that is, p< 2 N/ (N- 2) then there are at least two positive solutions for almost every 0 < λ< λ∗, the first one (that exists for all 0 < λ< λ∗) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every 0 < λ< λ∗) comes from an appropriate (and delicate) mountain pass argument. © 2017, Springer-Verlag GmbH Germany, part of Springer Nature.

Registro:

Documento: Artículo
Título:A concave–convex problem with a variable operator
Autor:Molino, A.; Rossi, J.D.
Filiación:Departamento de Análisis Matemático, Campus Fuentenueva S/N, Universidad de Granada, Granada, 18071, Spain
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina
Palabras clave:35J20; 35J62; 35J92
Año:2018
Volumen:57
Número:1
DOI: http://dx.doi.org/10.1007/s00526-017-1291-9
Título revista:Calculus of Variations and Partial Differential Equations
Título revista abreviado:Calc. Var. Partial Differ. Equ.
ISSN:09442669
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09442669_v57_n1_p_Molino

Referencias:

  • Ambrosetti, A., Brezis, H., Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems (1994) J. Funct. Anal., 122, pp. 519-543
  • Ambrosetti, A., Rabinowitz, P.H., Dual variational methods in critical point theory and applications (1973) J. Funct. Anal., 14, pp. 349-381
  • Ambrosetti, A., García-Azorero, J., Peral, I., Multiplicity results for some nonlinear elliptic equations (1996) J. Funct. Anal., 137, pp. 219-242
  • Acerbi, E., Fusco, N., A transmission problem in the calculus of variations (1994) Calc. Var. Partial. Differ. Equ., 2 (1), pp. 1-16
  • Ball, J.M., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations (1977) Q. J. Math., 28, pp. 473-486
  • Bulíček, M., Glitzky, A., Liero, M., Systems describing electrothermal effects with p (x) -Laplacian like structure for discontinuous variable exponents (2016) SIAM J. Math. Anal, 48 (5), pp. 3496-3514
  • Thermistor systems of p(x) -Laplace-type with discontinuous exponents via entropy solutions, , Bulíček, M., Glitzky, A., Matthias L.:(preprint MORE/2016/11)
  • Boccardo, L., Escobedo, M., Peral, I., A Dirichlet problem involving critical exponents (1995) Nonlinear Anal., 24 (11), pp. 1639-1648
  • Brezis, H., Nirenberg, L., H1 versus C1 local minimizers (1993) C. R. Acad. Sci. Paris, t.317, pp. 465-472
  • Charro, F., Colorado, E., Peral, I., Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave–convex right hand side (2009) J. Differ. Equ., 246, pp. 4221-4248
  • Damascelli, L., Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results (1998) Ann. Inst. H. Poincaré Anal. Nonlinéaire, 15, pp. 493-576
  • De Figueiredo, D.G., On the existence of multiple ordered solutions of nonlinear eigenvalue problems (1987) Nonlinear Anal. Theory Methods Appl., 11 (4), pp. 481-492
  • De Figueiredo, D.G., (1989) Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Published for the Tata Institute of Fundamental Research, Bombay, , Springer, Berlin
  • Diening, L., Harjulehto, P., Hästö, P., Ružička, M., (2017) Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, , Springer, Berlin
  • Fan, X.-L., Wang, S., Zhao, D., Density of C∞(Ω) in W1 , p ( x )(Ω) with discontinuous exponent p(x) (2006) Math. Nachr., 279, pp. 142-149
  • Fan, X.-L., Zhao, D., Regularity of quasi-minimizers of integral functionals with discontinuous p(x) -growth conditions (2006) Nonlinear Anal., 65 (8), pp. 1521-1531
  • Fischer, A., Koprucki, T., Gärtner, K., Brückner, J., Lüssem, B., Leo, K., Glitzky, A., Scholz, R., Feel the heat: nonlinear electrothermal feedback in organic LEDs (2014) Adv. Funct. Mater., 24, pp. 3367-3374
  • García-Azorero, J., Peral Alonso, I., Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term (1991) Trans. Am. Math. Soc., 323, pp. 877-895
  • García-Azorero, J., Peral Alonso, I., Some results about the existence of a second positive solution in a quasilinear critical problem (1994) Indiana Univ. Math. J., 43 (3), pp. 941-957
  • García-Azorero, J., Peral, I., Rossi, J.D., A convex–concave problem with a nonlinear boundary condition (2004) J. Differ. Equ., 198 (1), pp. 91-128
  • García-Azorero, J., Manfredi, J.J., Peral, I., Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations (2000) Commun. Contemp. Math., 2 (3), pp. 385-404
  • García-Melián, J., Rossi, J.D., Sabina de Lis, J.C., A variable exponent diffusion problem of concave–convex nature (2016) Topol. Methods Nonlinear Anal., 47 (2), pp. 613-639
  • García-Melián, J., Rossi, J.D., Sabina de Lis, J., A convex–concave elliptic problem with a parameter on the boundary condition (2012) Discrete Contin. Dyn. Syst., 32 (4), pp. 1095-1124
  • Gilbarg, D., Trudinger, N.S., (1983) Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 224. , 2, Springer, Berlin
  • Ghoussoub, N., Preiss, D., A general mountain pass principle for locating and classifying critical points (1989) Ann. Inst. H. Poincaré Anal. Non Linéaire, 6, pp. 321-330
  • Guedda, M., Veron, L., Quasilinear elliptic equations involving critical Sobolev exponents (1989) Nonlinear Anal. Theory Methods Appl., 13, pp. 879-902
  • Harjulehto, P., Hästö, P., Lê, Ú.V., Nuortio, M., Overview of differential equations with non-standard growth (2010) Nonlinear Anal., 72, pp. 4551-4574
  • Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesman–Lazer-type problem set on RN (1999) Proc. R. Soc. Edinb., 129A, pp. 787-809
  • Ladyzhenskaya, O.A., Ural’tseva, N.N., (1968) Linear and Quasilinear Elliptic Equations, , Academic Press, New York
  • Lions, P.L., On the existence of positive solutions of semilinear elliptic equations (1982) SIAM Rev., 24, pp. 441-467
  • Mercaldo, A., Rossi, J.D., Segura de León, S., Trombetti, C., On the behaviour of solutions to the Dirichlet problem for the p(x) -Laplacian when p(x) goes to 1 in a subdomain (2012) Differ. Integral Equ., 25 (1-2), pp. 53-74
  • Sakaguchi, S., Concavity properties of solutions to some degerate quasilinear elliptic Dirichlet Problems (1987) Ann. Scuola Normale Sup. di Pisa Serie 4, 14 (3), pp. 403-421
  • Stampacchia, G., (1966) Equations Elliptiques du Second Ordre a Coefficients Discontinus, , Les Presses de L’Université de Montreal, Montreal
  • Zhikov, V.V., On some variational problems (1997) Russ. J. Math. Phys., 8, pp. 105-116

Citas:

---------- APA ----------
Molino, A. & Rossi, J.D. (2018) . A concave–convex problem with a variable operator. Calculus of Variations and Partial Differential Equations, 57(1).
http://dx.doi.org/10.1007/s00526-017-1291-9
---------- CHICAGO ----------
Molino, A., Rossi, J.D. "A concave–convex problem with a variable operator" . Calculus of Variations and Partial Differential Equations 57, no. 1 (2018).
http://dx.doi.org/10.1007/s00526-017-1291-9
---------- MLA ----------
Molino, A., Rossi, J.D. "A concave–convex problem with a variable operator" . Calculus of Variations and Partial Differential Equations, vol. 57, no. 1, 2018.
http://dx.doi.org/10.1007/s00526-017-1291-9
---------- VANCOUVER ----------
Molino, A., Rossi, J.D. A concave–convex problem with a variable operator. Calc. Var. Partial Differ. Equ. 2018;57(1).
http://dx.doi.org/10.1007/s00526-017-1291-9