Nonlinear Analysis of PDE


The Nonlinear Analysis is one of the areas of Mathematics that has presented greater growth in the last years due to the fact that the methods developed there are adaptable to many others areas of Mathematics and to different types of Partial Differential Equations. These methods constitute a powerful tool to understand and to solve diverse problems coming from models in Biology, Finance and Control Theory.

Our group has special interest in studying fully nonlinear local and nonlocal elliptic and parabolic equations, which does not prevent us from dealing with problems that involve only the Laplace operator.
We are interested in proving existence and obtaining qualitative properties of the solutions for these types of equations. We have also a special interest in understanding the complexity of the solution sets that these equations sometimes hide. Notice that most of the qualitative properties and the structure of the solution sets that we are searching for will give rise to different interpretations from the theoretical and the applied point of views.

Members

  

Publications

  1. Quaas A, Xia A. Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions. Zeitschrift f{"u}r angewandte Mathematik und Physik 2016;67(3):1-21. [Digital version] [Bibtex]
  2. Quaas A, Xia A. A Liouville type theorem for Lane–Emden systems involving the fractional Laplacian. Nonlinearity 2016;29(8):2279. [Digital version] [Bibtex]
  3. Quaas A, García-Melián J, Alarcón S. Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian. Annali della Scuola Normale Superiore di Pisa. Classe di scienze 2016;16(1):129-158. [Digital version] [Bibtex]
  4. Quaas A, García-Melián J, Sirakov B. Elliptic equations with absorption in a half-space. Bulletin of the Brazilian Mathematical Society, New Series 2016:1-11. [Digital version] [Bibtex]
  5. Quaas A, Burgos-Pérez M. Á, García-Melián J. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete and Continuous Dynamical Systems 2016;36(9):4703-4721. [Digital version] [Bibtex]
  6. Barrios B, García-Melián J, Iturriaga L. Semilinear elliptic equations and nonlinearities with zeros. Nonlinear Anal. 2016;134:117-126. [Digital version] [Bibtex]
  7. Alarcón S, Burgos-Pérez M. Á, García-Melián J, Quaas A. Nonexistence results for elliptic equations with gradient terms. Journal of Differential Equations 2016;260:758-780. [Digital version] [Bibtex]
  8. Chen H, Felmer P, Quaas A. Large solutions to elliptic equations involving fractional Laplacian. Ann. Inst. Henri Poincare, Analyse non lineaire 2015;32(6):1199-1228. [Digital version] [Bibtex]
  9. Quaas A, Chen H, Felmer P. Self-generated interior blow-up solutions of fractional elliptic equation with absorption. Differential and Integral Equations 2015;28(9-10):839-860. [Digital version] [Bibtex]
  10. Quaas A, Xia A. Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space. Calculus of Variations and Partial Differential Equations 2015;52(3-4):641-659. [Digital version] [Bibtex]
  11. Alarcón S, Pistoia A. A Paneitz-type problem in pierced domains. Differential Integral Equations 2015;28:823-838. [Digital version] [Bibtex]
  12. Iturriaga L, García-Melián J, Quoirin H. R. A Priori Bounds and Existence of Solutions for Slightly Superlinear Elliptic Problems. Adv. Nonlinear Stud. 2015;15(3):923-938. [Digital version] [Bibtex]
  13. Iturriaga L, Dos Santos E. M, Ubilla P. Local minimizers in spaces of symmetric functions and applications. J. Math. Anal. Appl. 2015;429(1):27-56. [Digital version] [Bibtex]
  14. Flores I. Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball. Nonlinear Analysis: Theory, Methods & Applications 2015;125:128-149. [Digital version] [Bibtex]
  15. Flores I. Multiplicity results for the scalar curvature equation. Journal of Differential Equations 2015;259(8):4327-4355. [Digital version] [Bibtex]
  16. Topp E, Chasseigne E, Rossi J, Felmer P. Fractional Decay Bounds for Nonlocal Zero Order Heat Equations. Bull. Lond. Math. Soc. 2014;46(5):943–952. [Digital version] [Bibtex]
  17. Topp E. Existence and Uniqueness for Integro-Differential Equations with Dominating Drift Terms. Comm. Partial Differential Equations 2014;39(8):1523-1554. [Digital version] [Bibtex]
  18. Iturriaga L, Souto M. A, Ubilla P. Quasilinear Problems Involving Changing-Sign Nonlinearities Without an Ambrosetti–Rabinowitz-Type Condition. Proceedings of the Edinburgh Mathematical Society 2014;57(3):755-762. [Digital version] [Bibtex]
  19. Iturriaga L, Massa E, Sánchez J, Ubilla P. Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros. Mathematische Nachrichten 2014;287(10):1131-1141. [Digital version] [Bibtex]
  20. Alarcón S, García-Melián J, Quaas A. Existence and non-existence of solutions to elliptic equations with a general convection term. Proc. Roy. Soc. Edinburgh Sect. A 2014;144:225-239. [Digital version] [Bibtex]
  21. Topp E, Felmer P. Convergence Results for a Class of Nonlinear Fractional Heat Equations. Israel J. Math. 2013;198(1):1-34. [Digital version] [Bibtex]
  22. Quaas A, Alarcón S, Garcia-Mellian J. Liouville type theorems for elliptic equations with gradient terms.. Milan J. Math. 2013;81(1):171–185. [Digital version] [Bibtex]
  23. Quaas A, Felmer P, Sirakov B. Solvability of nonlinear elliptic equations with gradient terms.. J. Differential Equations 2013;254(11):4327–4346. [Digital version] [Bibtex]
  24. Flores I, Dávila J, Guerra I. MULTIPLICITY AND SINGULAR SOLUTIONS FOR A LIOUVILLE TYPE SYSTEM IN A BALL. Adv. in Differential Equations 2013;18(9-10):727-824. [Digital version] [Bibtex]
  25. Alarcón S, Diaz G, Rey J. M. The influence of sources terms on the boundary behavior of the large solutions of quasilinear elliptic equations. The power like case. Z. Angew. Math. Phys. 2013;64:659-677. [Digital version] [Bibtex]
  26. Alarcón S, Quaas A. Large viscosity solutions for some fully nonlinear equations. NoDEA Nonlinear Differential Equations Appl. 2013;20:1453-1472. [Digital version] [Bibtex]
  27. Alarcón S, Quaas A, García-Melián J. Nonexistence of positive supersolutions to some nonlinear elliptic problems. J. Math. Pures Appl. 2013;99:618-634. [Digital version] [Bibtex]
  28. Topp E, Dávila J. Concentrating Solutions of the Liouville Equation with Robin Boundary Condition. J. Differential Equations 2012;252(3):2648-2697. [Digital version] [Bibtex]
  29. Iturriaga L, Lorca S, Saavedra E, Ubilla P. Quasilinear equations involving nonlinear Neumann boundary conditions. Differential Integral Equations 2012;25(7-8):657-664. [Digital version] [Bibtex]
  30. Quaas A, Felmer P, Sirakov B. Existence and regularity results for fully nonlinear equations with singularities.. Math. Ann. 2012;354(1):377–400. [Digital version] [Bibtex]
  31. Quaas A, Felmer P. Boundary blow up solutions for fractional elliptic equations.. Asymptot. Anal. 2012;78(3):123–144. [Digital version] [Bibtex]
  32. Quaas A, Meneses R. Existence and non-existence of global solutions for uniformly parabolic equations. J. Evol. Equ. 2012;12(4):943–955. [Digital version] [Bibtex]
  33. Quaas A, Tan J, Felmer P. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian.. Proc. Roy. Soc. Edinburgh Sect. A 2012;142(6). [Digital version] [Bibtex]
  34. Alarcón S, Quaas A, García-Melián J. Existence and uniqueness of solutions to nonlinear elliptic equations without growth conditions at infinity. J. Anal. Math. 2012;118:83-104. [Digital version] [Bibtex]
  35. Alarcón S, Quaas A, Iturriaga L. Existence and multiplicity results for Pucci's operators involving nonlinearities with zeros. Cal. Var. 2012;45:443-454. [Digital version] [Bibtex]
  36. Tan J, Ying W, Jianfu Y. Nonlinear fractional field equations. Nonlinear Anal. 2012;75(4):2098–2110. [Digital version] [Bibtex]
  37. Alarcón S, Quaas A, García-Melián J. Keller-Osserman type conditions for some elliptic problems with gradient terms. Journal of Differential Equations 2012;252:886–914. [Digital version] [Bibtex]
  38. Quaas A, Felmer P. Fundamental solutions for a class of Isaacs integral operators.. Discrete Contin. Dyn. Syst. 2011;30(2):493–508. [Digital version] [Bibtex]
  39. Tan J, Jingang X. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete Contin. Dyn. Syst. 2011;31(3):975–983. [Digital version] [Bibtex]
  40. Tan J. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differential Equations 2011;42(1-2):21-41. [Digital version] [Bibtex]
  41. Allendes A, Quaas A. Multiplicity results for extremal operators through bifurcation. Discrete Continuous Dynamical Systems 2011;29(1):51-65. [Digital version] [Bibtex]
  42. Meneses R, Quaas A. Fujita type exponent for fully nonlinear parabolic equations and existence results. Journal of Mathematical Analysis and Applications 2011;376(2):514-527. [Digital version] [Bibtex]
  43. Felmer P, Quaas A. Fundamental solutions and Liouville type theorems for nonlinear integral operators. Advances in Mathematics 2011;226(3):2712-2738. [Digital version] [Bibtex]
  44. Iturriaga L, Massa E, Sánchez J, Ubilla P. Positive solutions of the {$p$}-{L}aplacian involving a superlinear nonlinearity with zeros. J. Differential Equations 2010;248(2):309-327. [Digital version] [Bibtex]
  45. Iturriaga L, Lorca S, Montenegro M. Existence of solutions to quasilinear elliptic equations with singular weights. Adv. Nonlinear Stud. 2010;10(1):109-120. [Digital version] [Bibtex]
  46. Iturriaga L, Lorca S, Massa E. Positive solutions for the {$p$}-{L}aplacian involving critical and supercritical nonlinearities with zeros. Ann. Inst. H. Poincaré Anal. Non Linéaire 2010;27(2):763-771. [Digital version] [Bibtex]
  47. Iturriaga L, Sánchez J. Exact number of solutions of stationary reaction-diffusion equations. Appl. Math. Comput. 2010;216(4):1250-1258. [Digital version] [Bibtex]
  48. Iturriaga L, Lorca S, Ubilla P. A quasilinear problem without the {A}mbrosetti-{R}abinowitz-type condition. Proc. Roy. Soc. Edinburgh Sect. A 2010;140(2):391-398. [Digital version] [Bibtex]
  49. Dávila J, Flores I, Guerra I. Multiplicity of solutions for a fourth order equation with power-type nonlinearity. 2010. [Digital version] [Bibtex]
  50. Davila G, Felmer P, Quaas A. Harnack inequality for singular fully nonlinear operators and some existence results. Calculus o Variations and PDE 2010;39(3-4):557-578. [Digital version] [Bibtex]
  51. Esteban M, Felmer P, Quaas A. Eigenvalues for Radially Symmetric Fully Nonlinear Operators. Communications in Partial Differential Equations 2010;35(9):1716 - 1737. [Digital version] [Bibtex]
  52. Quaas A, Felmer P, Sirakov B. Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations. Journal of Functional Analysis 2010;258:4154-4182. [Digital version] [Bibtex]
  53. Cabré X, Tan J. Positive solutions of nonlinear problems involving the square root of the Laplacian. Advances in Mathematics 2010;224(5):2052-2093. [Digital version] [Bibtex]
  54. Alarcón S, Díaz G, Letelier R, Rey J. M. Expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation. Nonlinear Analysis TMA 2010;72:2426-2443. [Digital version] [Bibtex]
  55. Esteban M, Felmer P, Quaas A. Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data. Proc. Roy. Soc. Edinburgh 2010;53:125-141. [Digital version] [Bibtex]
  56. Alarcón S, Quaas A. Large number of fast decay ground states to Matukuma-type equations. Journal of Differential Equations 2010;248:866-892. [Digital version] [Bibtex]
  57. Felmer P, Quaas A, Tan J. Geometry of phase plane and radial solutions for nonlinear elliptic equations with extremal operators. Journal Mathematical Analisys Aplication 2010;366(1):101-111. [Digital version] [Bibtex]
  58. Felmer P, Quaas A, Sirakov B. Resonance phenomena for second-order stochastic control equations. SIAM J. on Mathematical Analysis 2010;42(3):997-1024. [Digital version] [Bibtex]
  59. Dávila J, Flores I, Guerra I. Multiplicity of solutions for a fourth order equation with exponential nonlinearity. 2009. [Digital version] [Bibtex]
  60. Flores I. Singular solutions of the Brezis-Nirenberg problem in a ball. 2009. [Digital version] [Bibtex]
  61. Alarcón S. Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes. Commun. Pure Appl. Anal. 2009;8:1269-1289. [Digital version] [Bibtex]
  62. Felmer P, Quaas A. Fundamental solutions and two properties of elliptic maximal and minimal operators. Transactions of the American Mathematical Society 2009;361:5721-5736. [Digital version] [Bibtex]
  63. Felmer P, Quaas A, Tang M. On the complex structure of positive solutions to Matukuma-type equations. Ann. Inst. Henri Poincare, Analyse non lineaire 2009;26(3):869-887. [Digital version] [Bibtex]
  64. Quaas A, Sirakov B. Existence and non-existence results for fully nonlinear elliptic systems. ndiana Univ. Math Journal 2009;58(2). [Digital version] [Bibtex]
  65. Felmer P, Montenegro M, Quaas A. A note on the Strong Maximum Principle and the Compact Support Principle. Journal of Diff. Equation 2009;246(1):39-49. [Digital version] [Bibtex]
  66. Quaas A, Felmer P, Tang M, Yu J. Random dynamics of gene transcription activation in single cells. Journal of Differential Equations 2009;247(6):1796-1816. [Digital version] [Bibtex]
  67. Davila G, Felmer P, Quaas A. Alexandroff-Bakelman-Pucci estimate for singular or degenerate fully nonlinear elliptic equations. Comptes Rendus Mathematique, 2009;347(19-20):1165-1168. [Digital version] [Bibtex]
  68. Iturriaga L. Existence and multiplicity results for some quasilinear elliptic equation with weights. J. Math. Anal. Appl. 2008;339(2):1084-1102. [Digital version] [Bibtex]
  69. Brock F, Iturriaga L, Ubilla P. A multiplicity result for the {$p$}-{L}aplacian involving a parameter. Ann. Henri Poincaré 2008;9(7):1371-1386. [Digital version] [Bibtex]
  70. Iturriaga L, Lorca S, Sánchez J. Existence and multiplicity results for the {$p$}-{L}aplacian with a {$p$}-gradient term. NoDEA Nonlinear Differential Equations Appl. 2008;15(6):729-743. [Digital version] [Bibtex]
  71. Alarcón S. Double-spike solutions for a critical inhomogeneous elliptic problem in domains with small holes. Proc. Roy. Soc. Edinburgh Sect. A 2008;138:671-692. [Digital version] [Bibtex]
  72. Quaas A, Sirakov B. Principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Adv. in Mathematics 2008;218(1):105-135. [Digital version] [Bibtex]
  73. Felmer P, Quaas A, Tang M, Yu J. Monotonicity properties for ground states of the scalar field equation. Ann. Inst. Henri Poincare, Analyse non lineaire 2008;25(1):105-119. [Digital version] [Bibtex]
  74. Quaas A, Sirakov B. Solvability of monotone systems of fully nonlinear elliptic PDE´s. omptes Rendus Mathematique 2008;346(11-12):641-644. [Digital version] [Bibtex]
  75. Iturriaga L, Lorca S. Existence and multiplicity results for degenerate elliptic equations with dependence on the gradient. Bound. Value Probl. 2007:Art. ID 47218, 12. [Digital version] [Bibtex]
  76. Dobeault J, Flores I. Geometry of phase space and solutions of semi- linear elliptic equations in a ball. 2007. [Digital version] [Bibtex]
  77. Esteban M, Felmer P, Quaas A. Large critical exponents for some second order uniformly elliptic operators. Communications in Partial Differential Equations 2007;32(4):543 - 556. [Digital version] [Bibtex]
  78. Brock F, Iturriaga L, Ubilla P. Semi-linear singular elliptic equations with dependence on the gradient. Nonlinear Anal. 2006;65(3):601-614. [Digital version] [Bibtex]
  79. Brock F, Iturriaga L, Sánchez J, Ubilla P. Existence of positive solutions for {$p$}-{L}aplacian problems with weights. Commun. Pure Appl. Anal. 2006;5(4):941-952. [Digital version] [Bibtex]
  80. Felmer P, Quaas A. Critical exponents for uniformly elliptic extremal operators. Indiana Univ. Math Journal 2006;55(2):593-629. [Digital version] [Bibtex]
  81. Quaas A, Sirakov B. On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Comptes Rendus Mathematique 2006;342(2):115-118. [Digital version] [Bibtex]
  82. Quaas A, Sirakov B. Existence results for nonproper elliptic equation involving the Pucci´s Operator. Communications in Partial Differential Equations 2006;31(7):987 - 1003. [Digital version] [Bibtex]
  83. Felmer P, Quaas A, Tang M. On uniqueness for nonlinear elliptic equation involving the Pucci´s extremal operator. Journal of Diff. Equation 2006;226(1):80-98. [Digital version] [Bibtex]
  84. Busca J, Esteban M, Quaas A. Nonlinear Eigenvalues and Bifurcation problems for Pucci´s Operator. Ann. Inst. Henri Poincare, Analyse non lineaire 2005;22(2):187-206. [Digital version] [Bibtex]
  85. Flores I. A resonance phenomenon for ground states of an elliptic equation of Emden-Fowler type. 2004. [Digital version] [Bibtex]
  86. Quaas A. Existence of Positive Solutions to a ´semilinear´ equation involving the Pucci´s operator in a convex domain. Differential Integral Equation 2004;17(5-6):481-494. [Digital version] [Bibtex]
  87. Felmer P, Quaas A. Positive Radial Solutions to a ´semilinear´ equation involving the Pucci´s operator. J. Differential Equation 2004;199(2):376-393. [Digital version] [Bibtex]
  88. Felmer P, Quaas A. On Critical Exponents for Pucci´s Extremal Operators. Ann. Inst. Henri Poincare, Analyse non lineaire 2003;20(5):843-865. [Digital version] [Bibtex]
  89. Felmer P, Quaas A. Critical Exponents for Pucci´s Extremal Operators. C.R. Acad. Sci. Paris (I) 2002;335:909-914. [Digital version] [Bibtex]
  90. Felmer P, Quaas A. On The Strong Maximum Principle for Quasilinear Elliptic Equations and Systems. Adv. in Differential Equation 2002;7:25-46. [Digital version] [Bibtex]
  91. Busca J, Quaas A. Qualitative Properties for Semilinear Elliptic Systems with Non-Lipschitz Nonlinearity. Nonlinear Analysis TMA 2002;50(3):299-312. [Digital version] [Bibtex]
  92. Bamón R, Del Pino M, Flores I. Ground states of semilinear elliptic equations: a geometric approach. 2000. [Digital version] [Bibtex]
  93. Bamón R, Del Pino M, Flores I. Positive solutions of elliptic equations in R^N with a super-subcritical nonlinearity. 2000. [Digital version] [Bibtex]
  

Projects

  1. FONDECYT 1151180: Some Nonlinear local and non-local equations
  2. Millennium Nucleus Center for Analysis of PDE, NC130017
  3. CONCURSO “ATRACCIÓN DE CAPITAL HUMANO AVANZADO DEL EXTRANJERO” - Modalidad Estadías Cortas (MEC) –Consolidación de la línea de EDP's en el Doctorado Regional en Matemática
  4. MEC 80140118. Métodos Numéricos y Aplicaciones en el Doctorado Regional de Valparaíso.
  5. Mathamsud 13MATH-03 - QUESP - Quasilinear Equations and Singular Problems
  

Theses


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