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Combined effects in nonlinear problems arising in the study of anisotropic continuous media. (English) Zbl 1237.35043
This paper deals with the qualitative analysis of positive solutions for a class of nonlinear elliptic equations with Dirichlet boundary condition. The main features are the following: (i) the presence of variable potential functions; (ii) the study is performed provided that the nonlinear terms have subcritical growth and (possible) variable sign; (iii) the presence of a bifurcation parameter. By studying the competition between the terms arising in the equation, the authors establish several existence and nonexistence results, as well as an exhaustive bifurcation description. The proofs combine variational techniques with related estimates of the associated energy functional.
35J25Second order elliptic equations, boundary value problems
35B09Positive solutions of PDE
35B32Bifurcation (PDE)
35B40Asymptotic behavior of solutions of PDE
35J60Nonlinear elliptic equations
58E05Abstract critical point theory
[1]Gelfand, I. M.: Some problems in the theory of quasi-linear equations, Uspekhi mat. Nauk 14, 87-158 (1959)
[2]Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev. 18, 620-709 (1976) · Zbl 0345.47044 · doi:10.1137/1018114
[3]Keller, H. B.; Cohen, D. S.: Some positone problems suggested by nonlinear heat generation, J. math. Mech. 16, 1361-1376 (1967) · Zbl 0152.10401
[4]Mironescu, P.; Rădulescu, V.: A bifurcation problem associated to a convex, asymptotically linear function, C. R. Acad. sci. Paris sér. I 316, 667-672 (1993) · Zbl 0799.35025
[5]Mironescu, P.; Rădulescu, V.: The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear anal. 26, 857-875 (1996) · Zbl 0842.35008 · doi:10.1016/0362-546X(94)00327-E
[6]Cîrstea, F.; Ghergu, M.; Rădulescu, V.: Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane–Emden–Fowler type, J. math. Pures appl. 84, 493-508 (2005) · Zbl 1211.35111 · doi:10.1016/j.matpur.2004.09.005
[7]Whitham, G. B.: Linear and nonlinear waves, (1974) · Zbl 0373.76001
[8]Carleman, T.: Problèmes mathématiques dans la théorie cinétique des gaz, (1957) · Zbl 0077.23401
[9]Callegari, A.; Nachman, A.: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. math. 38, 275-281 (1980) · Zbl 0453.76002 · doi:10.1137/0138024
[10]Bandle, C.; Reichel, W.: Solutions of quasilinear second-order elliptic boundary value problems via degree theory, Handb. differ. Equ., 1-70 (2004) · Zbl 1129.35367
[11]Ghergu, M.; Rădulescu, V.: Singular elliptic problems: bifurcation and asymptotic analysis, Oxford lecture series in mathematics and its applications 37 (2008)
[12]Kristály, A.; Rădulescu, V.; Varga, C.: Variational principles in mathematical physics, geometry and economics: qualitative analysis of nonlinear equations and unilateral problems, Encyclopedia of mathematics 136 (2010)
[13]Rădulescu, V.: Qualitative analysis of nonlinear elliptic partial differential equations, Contemporary mathematics and its applications 6 (2008) · Zbl 1190.35003
[14]Ambrosetti, A.; Brezis, H.; Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems, J. funct. Anal. 122, 519-543 (1994) · Zbl 0805.35028 · doi:10.1006/jfan.1994.1078
[15]Alama, S.; Tarantello, G.: Elliptic problems with nonlinearities indefinite in sign, J. funct. Anal. 141, 159-215 (1996) · Zbl 0860.35032 · doi:10.1006/jfan.1996.0125
[16]P. Pucci, V. Rădulescu, Combined effects in quasilinear elliptic problems with lack of compactness, Rend. Lincei-Mat. Appl., in press (doi:10.4171/RLM).
[17]Wong, J. S. W.: On the generalized Emden–Fowler equation, SIAM rev. 17, 339-360 (1975) · Zbl 0295.34026 · doi:10.1137/1017036
[18]Lane, H.: On the theoretical temperature of the Sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Am. J. Sci. 50, 57-74 (1869)
[19]Keller, E. F.; Segel, L. A.: Initiation of slime mold aggregation viewed as an instability, J. theoret. Biol. 26, 399-415 (1970) · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[20]Gierer, A.; Meinhardt, H.: A theory of biological pattern formation, Kybernetik 12, 30-39 (1972)
[21]De Gennes, P. G.: Wetting: statics and dynamics, Rev. modern phys. 57, 827-863 (1985)
[22]Chayes, J. T.; Osher, S. J.; Ralston, J. V.: On singular diffusion equations with applications to self-organized criticality, Comm. pure appl. Math. 46, 1363-1377 (1993) · Zbl 0832.35142 · doi:10.1002/cpa.3160461004
[23]Vázquez, J. L.: Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. math. Pures appl. 71, 503-526 (1992) · Zbl 0694.35088
[24]Bartsch, T.; Willem, M.: On an elliptic equation with concave and convex nonlinearities, Proc. amer. Math. soc. 123, 3555-3561 (1995) · Zbl 0848.35039 · doi:10.2307/2161107
[25]Il’yasov, Y.: On nonlocal existence results for elliptic equations with convex–concave nonlinearities, Nonlinear anal. 61, 211-236 (2005) · Zbl 1190.35112 · doi:10.1016/j.na.2004.10.022
[26]Lubyshev, V. F.: Multiple positive solutions of an elliptic equation with a convex–concave nonlinearity containing a sign-changing term, Tr. mat. Inst. steklova 269, 167-180 (2010) · Zbl 1202.35111 · doi:10.1134/S0081543810020148
[27]Lubyshev, V. F.: Multiple solutions of an even order nonlinear problem with convex–concave nonlinearity, Nonlinear anal. 74, 1345-1354 (2011) · Zbl 1205.35093 · doi:10.1016/j.na.2010.10.007
[28]Pohozaev, S. I.: On an approach to nonlinear equations, Dokl. acad. Sci. USSR 247, 1327-1331 (1979)
[29]Bozhkov, Y.; Mitidieri, E.: Existence of multiple solutions for quasilinear systems via fibering method, J. differential equations 190, 239-267 (2003) · Zbl 1021.35034 · doi:10.1016/S0022-0396(02)00112-2
[30]Bozhkov, Y.; Mitidieri, E.: Existence of multiple solutions for quasilinear equations via fibering method, Progr. nonlinear differential equations appl. 66, 115-134 (2006) · Zbl 1131.35319
[31]Azorero, J. Garcia; Alonso, I. Peral: Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. amer. Math. soc. 323, 877-895 (1991) · Zbl 0729.35051 · doi:10.2307/2001562
[32]Brezis, H.; Kamin, S.: Sublinear elliptic equations in rn, Manuscripta math. 74, 87-106 (1992) · Zbl 0761.35027 · doi:10.1007/BF02567660
[33]Pucci, P.; Serrin, J.: The strong maximum principle revisited, J. differential equations 196, 1-66 (2004) · Zbl 1109.35022 · doi:10.1016/j.jde.2004.09.002
[34]Filippucci, R.; Pucci, P.; Rădulescu, V.: Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. partial differential equations 33, 706-717 (2008) · Zbl 1147.35038 · doi:10.1080/03605300701518208
[35]Pucci, P.; Servadei, R.: On weak solutions for p-Laplacian equations with weights, Rend. lincei-mat. Appl. 18, 257-267 (2007) · Zbl 1223.35128 · doi:10.4171/RLM/493
[36]Pucci, P.; Serrin, J.: On the strong maximum and compact support principles and some applications, Handbook of differential equations–stationary partial differential equations 4, 355-483 (2007)