Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities

Abstract In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane—Emden—Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convection nonlinearities, as well as sign-changing potentials. We also take into account bifurcation nonlinear problems and we establish the precise rate decay of the solution in some concrete situations. Our approach combines standard techniques based on the maximum principle with nonstandard arguments, such as the Karamata regular variation theory.

[1]  Anthony Joseph,et al.  First European Congress of Mathematics , 1994 .

[2]  Vicentiu D. Rădulescu,et al.  Asymptotics for the Blow-Up Boundary Solution of the Logistic Equation with Absorption , 2003 .

[3]  Yihong Du,et al.  Blow-Up Solutions for a Class of Semilinear Elliptic and Parabolic Equations , 1999, SIAM J. Math. Anal..

[4]  Alexander Meadows.  Stable and singular solutions of the equation Δu = 1/u , 2004 .

[5]  M. Pierre,et al.  Weak solutions of some quasilinear elliptic equations with data measures , 1993 .

[6]  Charles Alexander Stuart,et al.  Existence and approximation of solutions of non-linear elliptic equations , 1976 .

[7]  R. Redheffer.  On the inequality Δu⩾f(u,¦grad u¦) , 1960 .

[8]  Vicentiu D. Rădulescu,et al.  Existence and nonexistence of entire solutions to the logistic differential equation , 2003 .

[9]  Vicenţiu D. Rădulescu,et al.  EXISTENCE AND UNIQUENESS OF BLOW-UP SOLUTIONS FOR A CLASS OF LOGISTIC EQUATIONS , 2002 .

[10]  P. Gennes.  Wetting: statics and dynamics , 1985 .

[11]  Florica-Corina St . Cand.  SOLUTIONS WITH BOUNDARY BLOW-UP FOR A CLASS OF NONLINEAR ELLIPTIC PROBLEMS , 2003 .

[12]  L. Véron,et al.  Scalar Curvature and Conformal Deformation of Hyperbolic Space , 1994 .

[13]  Y. Haitao,et al.  Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem , 2003 .

[14]  Nonradial blow-up solutions of sublinear elliptic equations with gradient term , 2004, math/0502143.

[15]  L. Bieberbach.  Δu=eu und die automorphen Funktionen , 1916 .

[16]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[17]  Jorge García-Melián,et al.  Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up , 2001 .

[18]  Vicentiu D. Rădulescu,et al.  Singular elliptic problems with convection term in anisotropic media , 2006, math/0606165.

[19]  C. Bandle,et al.  Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curvature of the boundary , 2004 .

[20]  Junping Shi,et al.  Positive solutions for elliptic equations with singular nonlinearity. , 2005 .

[21]  M. Crandall.  A Semilinear Equation in L1(rn) , 2007 .

[22]  H. Amann.  Existence and multiplicity theorems for semi-linear elliptic boundary value problems , 1976 .

[23]  Jesús Ildefonso Díaz Díaz,et al.  An elliptic equation with singular nonlinearity , 1987 .

[24]  Vicentiu D. Rădulescu,et al.  Multi-parameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[25]  Zhijun Zhang,et al.  On a Dirichlet Problem with a Singular Nonlinearity , 1995 .

[26]  F. W. Warner,et al.  Remarks on some quasilinear elliptic equations , 1975 .

[27]  J. Serrin,et al.  CHAPTER 6 - Maximum Principles for Elliptic Partial Differential Equations , 2007 .

[28]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[29]  Vicentiu D. Rădulescu,et al.  Sublinear singular elliptic problems with two parameters , 2003 .

[30]  P. Quittner.  Blow-up for semilinear parabolic equations with a gradient term , 1991 .

[31]  J. Karamata,et al.  Sur un mode de croissance régulière. Théorèmes fondamentaux , 1933 .

[32]  J. Matero.  Quasilinear elliptic problems with boundary blow-up , 1995 .

[33]  Vicentiu D. Rădulescu,et al.  Extremal singular solutions for degenerate logistic-type equations in anisotropic media , 2004 .

[34]  Vicentiu D. Rădulescu,et al.  The study of a bifurcation problem associated to an asymptotically linear function , 1996 .

[35]  A. Wood,et al.  Large solutions of semilinear elliptic equations with nonlinear gradient terms , 1999 .

[36]  Corina Cîrstea,et al.  Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach , 2005, Asymptot. Anal..

[37]  H. Brezis,et al.  Elliptic and parabolic problems : a special tribute to the work of Haim Brezis , 2005 .

[38]  Michel Chipot,et al.  Elements of Nonlinear Analysis , 2000 .

[39]  Yoshikazu Nakai,et al.  Some remarks on strongly invariant rings , 1975 .

[40]  Vicentiu D. R ˘ adulescu.  EXISTENCE AND UNIQUENESS OF BLOW-UP SOLUTIONS FOR A CLASS OF LOGISTIC EQUATIONS , 2002 .

[41]  V. D. Rù.  Entire solutions blowing up at infinity for semilinear elliptic systems , 2002 .

[42]  Zhijun Zhang,et al.  On a Singular Nonlinear Dirichlet Problem with a Convection Term , 2000, SIAM J. Math. Anal..

[43]  Y. Choi,et al.  Some remarks on a singular elliptic boundary value problem , 1998 .

[44]  Catherine Bandle,et al.  Boundary blow up for semilinear elliptic equations with nonlinear gradient terms , 1996, Advances in Differential Equations.

[45]  Robert Osserman,et al.  On the inequality $\Delta u\geq f(u)$. , 1957 .

[46]  Vicentiu D. Rădulescu.  Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity , 2005, math/0502142.

[47]  Vicentiu D. Rădulescu,et al.  On a class of sublinear singular elliptic problems with convection term , 2005, math/0504593.

[48]  Haim Brezis,et al.  Remarks on sublinear elliptic equations , 1986 .

[49]  L. Nirenberg,et al.  Partial Differential Equations Invariant under Conformal or Projective Transformations , 1974 .

[50]  Manuel del Pino,et al.  A global estimate for the gradient in a singular elliptic boundary value problem , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[51]  Victor A. Galaktionov,et al.  The problem of blow-up in nonlinear parabolic equations , 2002 .

[52]  S. Herminghaus,et al.  Wetting: Statics and dynamics , 1997 .

[53]  Vicentiu D. Rădulescu,et al.  Lane-Emden-Fowler equations with convection and singular potential , 2007 .

[54]  Thomas Mikosch,et al.  Regularly varying functions , 2006 .

[55]  C. Stuart.  Self-trapping of an electromagnetic field and bifurcation from the essential spectrum , 1991 .

[56]  G. Rosen.  The mathematical theory of diffusion and reaction in permeable catalysts , 1976 .

[57]  M. Marcus.  On Solutions with Blow-Up at the Boundary for a Class of Semilinear Elliptic Equations , 1992 .

[58]  Laurent Véron,et al.  Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations , 1997 .

[59]  M. M. Coclite,et al.  On a singular nonlinear dirichlet problem , 1989 .

[60]  Vicenţiu D. Rădulescu,et al.  Uniqueness of the blow-up boundary solution of logistic equations with absorbtion , 2002 .

[61]  Constantin P. Niculescu,et al.  Explosive solutions of elliptic equations with absorption and nonlinear gradient term , 2002 .

[62]  Marius Ghergu,et al.  Explosive solutions of semilinear elliptic systems with gradient term. , 2003 .

[63]  Alan C. Lazer,et al.  Asymptotic behavior of solutions of boundary blowup problems , 1994, Differential and Integral Equations.

[64]  Hongwei Chen.  On a singular nonlinear elliptic equation , 1997 .

[65]  Vicentiu D. Rădulescu,et al.  Blow-up boundary solutions of semilinear elliptic problems , 2002 .

[66]  P. J. McKenna,et al.  On a problem of Bieberbach and Rademacher , 1993 .

[67]  M. Meerschaert.  Regular Variation in R k , 1988 .

[68]  J. Vega,et al.  On the linearization of some singular, nonlinear elliptic problems and applications , 2002 .

[69]  Patrizia Pucci,et al.  The strong maximum principle revisited , 2004 .

[70]  A variational problem related to self-trapping of an electromagnetic field , 1996 .

[71]  R. Dalmasso.  Solutions d'équations elliptiques semi-linéaires singulières , 1988 .

[72]  J. Keller.  On solutions of δu=f(u) , 1957 .

[73]  J. Gall.  A Path-Valued Markov Process and its Connections with Partial Differential Equations , 1994 .

[74]  Haim Brezis,et al.  Sublinear elliptic equations in ℝn , 1992 .

[75]  M Gomes Sônia,et al.  On a singular nonlinear elliptic problem , 1986 .

[76]  S. Alama,et al.  On the solvability of a semilinear elliptic equation via an associated eigenvalue problem , 1996 .

[77]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[78]  Changfeng Gui,et al.  Regularity of an elliptic problem with a singular nonlinearity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[79]  A. Callegari,et al.  A Nonlinear Singular Boundary Value Problem in the Theory of Pseudoplastic Fluids , 1980 .

[80]  P. Lions,et al.  Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints , 1989 .

[81]  Nonlinear Singular Elliptic Problems: Recent Results and Open Problems , 2005 .

[82]  Stable and singular solutions of the equation $\Delta u = 1/u$ , 2004, math/0404422.

[83]  Vicentiu D. Rădulescu,et al.  Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type , 2005 .

[84]  Zhijun Zhang.  Nonexistence of positive classical solutions of a singular nonlinear Dirichlet problem with a convection term , 1996 .

[85]  R. Hardt,et al.  Minimal surfaces with isolated singularities , 1984 .

[86]  Vicenţiu D. Rădulescu,et al.  Bifurcation and asymptotics for the Lane–Emden–Fowler equation , 2003 .

[87]  Catherine Bandle,et al.  Asymptotic behaviour of large solutions of quasilinear elliptic problems , 2003 .

[88]  L. M. Berkovich.  The Generalized Emden-Fowler Equation , 1997 .

[89]  Vicenţiu D. Rădulescu,et al.  Entire solutions blowing up at infinity for semilinear elliptic systems , 2002 .

[90]  A. U.S..  On a singular nonlinear semilinear elliptic problem , 1998 .

[91]  Vicentiu D. Rădulescu,et al.  Bifurcation for a class of singular elliptic problems with quadratic convection term , 2004 .

[92]  Alan C. Lazer,et al.  On a singular nonlinear elliptic boundary-value problem , 1991 .

[93]  A. Callegari,et al.  Some singular, nonlinear differential equations arising in boundary layer theory , 1978 .

[94]  G. Barles,et al.  Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non lipschitz nonlinearity , 1992 .

[95]  Laurent Veron,et al.  Singularities of solutions of second order quasilinear equations , 1996 .

[96]  Vicentiu D. Rădulescu,et al.  Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type , 2005, math/0506121.

[97]  Jesús Ildefonso Díaz Díaz,et al.  Nonlinear Partial Differential Equations , 2012 .

[98]  Alan V. Lair,et al.  Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems , 2000 .

[99]  G. Temple.  Partial Differential Equations of Elliptic Type , 1971 .