Abstract:
We consider positive solutions to the non-variational family of Equations-△u-b(x)|∇u|2=λg(u) in Ω where λ ≥ 0, b(x) is a given function, g is an increasing nonlinearity with g(0) > 0 and Ω ℝn is a bounded smooth domain. We introduce the definition of stability for non-variational problems and establish existence and regularity results for stable solutions. These results generalize the classical results obtained when b(x) = b is a constant function making the problem variational after a suitable transformation. © 2016 Texas State University.
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Documento: |
Artículo
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Título: | Stable solutions for equations with a quadratic gradient term |
Autor: | Terra, J. |
Filiación: | Departamento de Matemática, Fceyn, and Imas-Conicet, UBA, Buenos Aires, 1428, Argentina
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Palabras clave: | Non-variational problem; Stable solution |
Año: | 2016
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Volumen: | 2016
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Título revista: | Electronic Journal of Differential Equations
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Título revista abreviado: | Electron. J. Differ. Equ.
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ISSN: | 10726691
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10726691_v2016_n_p_Terra |
Referencias:
- Abdellaoui, B., Dall’Aglio, A., Peral, I., Some remarks on elliptic problems with critical growth in the gradient (2006) J. Diff. Equations, 1, pp. 21-62
- Alama, S., Bronsard, L., Gui, C., Stationary layered solutions in R2 for an Allen-Cahn system with multiple well potential (1997) Calc. Var. Partial Differential Equations, 5, pp. 359-390
- Alberti, G., Ambrosio, L., Cabre, X., On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property (2001) Acta Appl. Math, 65, pp. 9-33
- Alencar, H., Barros, A., Palmas, O., Reyes, J.G., Santos, W., O(M) x O(n)-invariant minimal hypersurfaces in Rm+n (2005) Ann. Global Anal. Geom, 27, pp. 179-199
- Alessio, F., Calamai, A., Montecchiari, P., Saddle-type solutions for a class of semilinear elliptic equations, Adv (2007) Differential Equations, 12, pp. 361-380
- Almgren, F.J., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem (1966) Ann. of Math., 85, pp. 277-292
- Amann, H., Crandall, M.G., On Some Existence Theorems for Semi-linear Elliptic Equations (1978) Ind. Univ. Math. J., 27 (5), pp. 779-790
- Ambrosio, L., Cabre, X., Entire Solutions of Semilinear Elliptic Equations in R3 and a Conjecture of De Giorgi (2000) Journal Amer. Math. Soc, 13, pp. 725-739
- Berestycki, H., Cabre, X., Ryzhik, L., Bounds for the Explosion Problem in a Flow, , preparation
- Berestycki, H., Caffarelli, L., Nirenberg, L., Further qualitative properties for elliptic equations in unbounded domains (1997) Ann. Scuola Norm. Sup. Pisa Cl. Sci, 25, pp. 69-94
- Berestycki, H., Hamel, F., Nadirashvili, N., The speed of propagation for KPP type problems (2005) I. Periodic Framework, J. Eur. Math. Soc., 7 (2), pp. 173-221
- Berestycki, H., Hamel, F., Rossi, L., Louville type results for semilinear elliptic equations in unbounded domains Ann. Mat. Pura Appl.
- Berestycki, H., Nirenberg, L., Varadhan, S.R.S., The Principal Eigenvalue and Maximum Principle for Second-Order Elliptic Operators in General Domains (1994) Comm. Pure Appl. Math, 47, pp. 47-92
- Boccardo, L., T-minima: An approach to minimization problems in L1, Contributi dedicati alla memoria di Ennio De Giorgi (2000) Rich. Mat, 49, pp. 135-154
- Bombieri, E., De Giorgi, E., Giusti, E., Minimal cones and the Bernstein problem (1969) Inv. Math, 7, pp. 243-268
- Brezis, H., Cabre, X., Some simple nonlinear PDE's without solutions (1998) Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (1-2), pp. 223-262
- Brezis, H., Vazquez, J.L., Blow-up solutions of some nonlinear elliptic problems (1997) Rev. Mat. Univ. Comp. Madrid, 10 (2), pp. 443-468
- Cabre, X., Boundedness of Minimizers of Semilinear Problems up to Dimension Four
- Cabré, X., Capella, A., On the stability of radial solutions of semilinear elliptic equations in all of Rn (2004) C. R. Acad. Sci. Paris, Ser, 1 (338), pp. 769-774
- Cabre, X., Capella, A., Regularity of radial minimizers and extremal solutions of semilinear elliptic equations (2006) J. Functional Analysis, 238, pp. 709-733
- Cabré, X., Sanchon, M., Spruck, J., A priori estimates for semistable solutions of semilinear elliptic equations (2016) Dis. Cont. Dyn. Systems, 362, pp. 601-609
- Cabré, X., Terra, J., Saddle-shaped solutions of bistable diffusion equations in all of R2m (2009) Jour. of the European Math. Society, 11, pp. 819-843
- Cabré, X., Terra, J., Qualitative properties of saddle-shaped solutions to bistable diffusion equations (2010) Comm. in Partial Differential Equations, 35, pp. 1923-1957
- Caffarelli, L., Garofalo, N., Segaia, F., A gradient bound for entire solutions of quasi-linear equations and its consequences (1994) Comm. Pure and Appl. Math, 47 (11), pp. 1457-1473
- Castorina, D., Sanchon, M., Regularity of stable solutions to semilinear elliptic equations on Riemannian models (2015) Adv. Nonlinear Anal, 4 (4), pp. 295-309
- Cirstea, F., Ghergu, M., Radulescu, V., Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type (2005) J. Math. Pures Appliquees, 84, pp. 493-508
- Crandall, M.G., Rabinowitz, P.H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems (1975) Arch. Rat. Mech. Anal., 58
- Dancer, E.N., Farina, A., On the classification of solutions of An = eu on: Stability outside a compact set and applications (2009) Proc. Amer. Math. Soc, 137, pp. 1333-1338
- Dang, H., Fife, P.C., Peletier, L.A., Saddle solutions of the bistable diffusion equation (1992) Z. Angew Math. Phys, 43 (6), pp. 984-998
- Davila, J., Dupaigne, L., Perturbing singular solutions of the Gelfand problem (2007) Commun. Contemp. Math, 9 (5), pp. 639-680
- De Giorgi, E., Una estensione del teorema di Bernstein (1965) Ann. Sc. Norm. Sup. Pisa, 19, pp. 79-85
- De Giorgi, E., Convergence problems for functionals and operators (1979) Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, pp. 131-188. , Pitagora, Bologna
- Dupaigne, L., Farina, A., Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities (2009) Nonlinear Anal, 70 (8), pp. 2882-2888
- Esposito, P., Linear Unstability of Entire Solutions for a Class Ot Non-Autonomous Elliptic Equations
- Farina, A., On the classification of solutions of the Lane-Emden equation on unbounded domains of RN (2007) J. Math. Pures Appl, 87, pp. 537-561
- Farina, A., Stable solutions of-An = eu on RN (2007) C. R. Math. Acad. Sci. Paris, 345, pp. 63-66
- Fleming, W.H., On the oriented Plateau problem, Rend. Circolo Mat (1962) Palermo, 9, pp. 69-89
- Ghergu, M., Radulescu, V., Singular Elliptic Problems. Bifurcation and Asymptotic Analysis (2008) Oxford Lecture Series in Mathematics and Its Applications, 37. , Oxford University Press
- Ghoussoub, N., Gui, C., On a conjecture of De Giorgi and some related problems (1998) Math. Ann, 311, pp. 481-491
- Giusti, E., (1984) Minimal Surfaces and Functions of Bounded Variation, , Birkhauser Verlag, BaselBoston
- Jerison, D., Monneau, R., Towards a counter-example to a conjecture of De Giorgi in high dimensions (2004) Ann. Mat. Pura Appl., 183 (4), pp. 439-467
- Joseph, D.D., Lundgren, T.S., Quasilinear Dirichlet problems driven by positive sources (1973) Arch. Ration. Mech. Anal., 49, pp. 241-268
- Kardar, M., Parisi, G., Zhang, Y.C., Dynamic scaling of growing interfaces, Phys (1986) Rev. Lett., 56, pp. 889-892
- Ladyzhenskaya, O.A., Ural’Ceva, N.N., (1968) Linear and Quasi-Linear Elliptic Equations, , Academic Press, New York-London
- Leray, J., Lions, J.-L., Quelques résultats de Visik sur les problemes elliptiques non linéaires par les méthodes de Minty-Browder (1965) Bull. Soc. Math. France, 93, pp. 97-107
- Lions, P.L., Generalized solutions of Hamilton-Jacobi equations (1982) Pitman Research Notes in Math., Vol., p. 62
- Modica, L., A gradient bound and a Liouville theorem for nonlinear Poisson equations (1985) Comm. Pure Appl. Math, 38, pp. 679-684
- Modica, L., Monotonicity of the energy for entire solutions of semilinear elliptic equations (1989) Partial Differential Equations and the Calculus of Variations, 2-2, pp. 843-850. , Progr. Nonlinear Differential Equations Appl, Birkhauser, Boston
- Modica, L., Mortola, S., Un esempio di r-convergenza (1977) Boll. Un. Mat. Ital. B, 14, pp. 285-299
- Mignot, F., Puel, J.-P., Sur une classe de problemes non linéaires avec nonlinéarité positive, croissante, convexe (1980) Comm. Partial Differential Equations, 5, pp. 791-836
- Mironescu, P., Radulescu, V., A bifurcation problem associated to a convex, asymptotically linear function (1993) C.R. Acad. Sci. Paris, Ser, 1 (316), pp. 667-672
- Mironescu, P., Radulescu, V., The study of a bifurcation problem associated to an asymptotically linear function (1996) Nonlinear Analysis, T.M.A., 26 (4), pp. 857-875
- Nedev, G., Regularity of the extremal solution of semilinear elliptic equations (2000) C. R. Acad. Sci. Paris, Ser. I Math, 330, pp. 997-1002
- Savin, O., Regularity of flat level sets in phase transitions (2009) Ann. of Math, 169 (1), pp. 41-78
- Schatzman, M., On the stability of the saddle solution of Allen-Cahn’s equation (1995) Proc. Roy. Soc. Edinburgh Sect. A, 125 (6), pp. 1241-1275
- Simons, J., Minimal varieties in riemannian manifolds (1968) Ann. Math, 88, pp. 62-105
- Shi, J., Saddle solutions of the balanced bistable diffusion equation (2002) Comm. Pure. Appl. Math, 55 (7), pp. 815-830
- Villegas, S., Boundedness of extremal solutions in dimension 4 (2013) Adv. Math, 235, pp. 126-133
- Villegas, S., Asymptotic behavior of stable radial solutions of semilinear elliptic equations in Rn (2007) J. Math. Pures Appl, 88 (3), pp. 241-250
Citas:
---------- APA ----------
(2016)
. Stable solutions for equations with a quadratic gradient term. Electronic Journal of Differential Equations, 2016.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10726691_v2016_n_p_Terra [ ]
---------- CHICAGO ----------
Terra, J.
"Stable solutions for equations with a quadratic gradient term"
. Electronic Journal of Differential Equations 2016
(2016).
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10726691_v2016_n_p_Terra [ ]
---------- MLA ----------
Terra, J.
"Stable solutions for equations with a quadratic gradient term"
. Electronic Journal of Differential Equations, vol. 2016, 2016.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10726691_v2016_n_p_Terra [ ]
---------- VANCOUVER ----------
Terra, J. Stable solutions for equations with a quadratic gradient term. Electron. J. Differ. Equ. 2016;2016.
Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10726691_v2016_n_p_Terra [ ]