Existence, non-existence and asymptotic behaviour of blow-up entire solutions of quasi-linear elliptic equations

References

• Guedda , M and Veron , L . 1988 . Local and global properties of solutions of quasilinear elliptic equations . J. Differential Equations , 76 : 159 – 189 .
   Web of Science ®Google Scholar
• Astrita , G and Marrucci , G . 1974 . Principles of non-Newtonian fluid mechanics , New York : McGraw-Hill .
   Google Scholar
• Martinson , KL and Pavlov , KB . 1971 . Unsteady shear flows of a conducting fluid with a rheological power law . Magnitnaya Gidrodinamika , 2 : 50 – 58 .
   Google Scholar
• Kalashnikov , AS . 1978 . On a nonlinear equation appearing in the theory of non-stationary filtration . Trud. Sem. I.G. Petrovski , 4 : 137 – 146 . [in Russian]
   Google Scholar
• Esteban , JR and Vazquez , JL . 1982 . On the equation of turbulent filteration in one-dimensional porous media . Nonlinear Anal. , 10 : 1303 – 1325 .
   Web of Science ®Google Scholar
• Guo , Z . 1992 . Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems . Nonlinear Anal. , 18 : 957 – 971 .
   Web of Science ®Google Scholar
• Guo , Z and Webb , JRL . 1994 . Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large . Proc. Royal Soc. Edinburgh , 124A : 189 – 198 .
   Google Scholar
• Guo , Z . 1996 . On the number of positive solutions for quasilinear elliptic eigenvalue problems . Nonlinear Anal. , 27 ( 2 ) : 229 – 247 .
   Web of Science ®Google Scholar
• Guo , Z . 1992 . Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations . Applicable Anal , 47 : 173 – 190 .
   Google Scholar
• Guo , Z . 1993 . Boundary value problems of a class of quasilinear ordinary differential equations . Differential Integral Equations. , 6 ( 3 ) : 705 – 719 .
   Google Scholar
• Guo , Z and Yang , Z . 1998 . Some uniqueness results for a class of quasilinear elliptic eigenvalue problems . Acta Mathematica Sinica (new series) , 14 ( 2 ) : 245 – 260 .
   Google Scholar
• Pino , MD , Elgueta , M and Manasevich , R . 1989 . A homotopic deformation along p of a Laray-Schauder degree result and existence for (|u′| p−2 u′)′ + f(t, u) = 0, u(0) = u(T) = 0, p > 1 . J. Differential Equations , 80 : 1 – 13 .
   Web of Science ®Google Scholar
• Sakaguchi , S . 1987 . Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) , 14 : 403 – 421 .
   Google Scholar
• Yang , Z and Guo , Z . 1995 . On the structure of positive solutions for quasilinear ordinary differential equations . Appl. Anal. , 58 : 31 – 51 .
   Google Scholar
• Yang , Z and Yang , H . 2001 . A priori for a quasilinear elliptic P.D.E. non-positone problems . Nonlinear Anal. , 43 ( 2 ) : 173 – 181 .
   Web of Science ®Google Scholar
• Yang , Z . 2004 . Existence of positive bounded entire solutions for quasilinear elliptic equations . Appl. Math. Comput. , 156 : 743 – 754 .
   Web of Science ®Google Scholar
• Hai , DD . 1998 . Positive solutions of quasilinear boundary value problems . J. Math. Anal. Appl. , 217 : 672 – 686 .
   Web of Science ®Google Scholar
• Benjin , X and Zuchi , C . 1999 . Solvability of singular quasilinear elliptic equation . Chinese Ann of Math. , 20A ( 1 ) : 117 – 128 .
   Google Scholar
• Yang , Z . 2006 . Existence of Positive Entire Solutions for Singular and Non-Singular Quasi-linear Elliptic Equation . J. Comm. Appl. Math. , 197 : 355 – 364 .
   Web of Science ®Google Scholar
• Zhang , Z . 1995 . On a Dirichlet problem with singular nonlinearity . J. Math. Anal. Appl. , 194 ( 1 ) : 103 – 113 .
   Google Scholar
• Diaz , JI , Morel , JM and Oswald , L . 1987 . An elliptic equation with singular nonlinearity . Comm. Partial Differential Equations , 12 ( 12 ) : 1333 – 1344 .
   Web of Science ®Google Scholar
• Lair , AV and Shaker , AW . 1996 . Entire solution of a singular semilinear elliptic problem . J. Math. Anal. Appl. , 200 : 498 – 505 .
   Web of Science ®Google Scholar
• Lazer , AC and McKenna , PJ . 1991 . On a singular nonlinear elliptic boundary value problem . Proc. Amer. Math. Soc. , 111 ( 3 ) : 721 – 730 .
   Web of Science ®Google Scholar
• Edelson , A . 1989 . Entire solutions of singular elliptic equations . J. Math. Anal. Appl. , 139 : 523 – 532 .
   Web of Science ®Google Scholar
• Kusano , T and Swanson , CA . 1985 . Entire positive solutions of singular semilinear elliptic equations . Japan J. Math. , 11 : 145 – 155 .
   Google Scholar
• Jin , Z . 1998 . Solutions for a class of singular semilinear elliptic equations . Nonlinear Analysis , 31 : 475 – 492 .
   Google Scholar
• Miaoxin , Y . 1995 . Positive Entire Solutions To Singular Semilinear Elliptic Equations Of Mixed Type . Transactions of Tianjin University , 1 ( 1 ) : 38 – 41 .
   Google Scholar
• Junping , S and Miaoxin , Y . 1998 . On a singular nonlinear semilinear elliptic problem . Proc. Roy. Soc. Edinburgh Sect. A , 128 : 1398 – 1401 .
   Google Scholar
• Maagli , H and Zribi , M . 2001 . Existence and estiments of solutions for singular nonlinear elliptic problems . J. Math. Anal. Appl. , 263 : 522 – 542 .
   Web of Science ®Google Scholar
• Anuradha , V , Brown , C and Shivaji , R . 1996 . Explosive nonnegative solutions to two point boundary value problems . Nonlinear Anal. , 26 : 613 – 630 .
   Google Scholar
• Wang , SH . 2000 . Existence and multiplicity of boundary blow-up nonnegative solutions to two-point boundary value problems . Nonlinear Anal. , 42 : 139 – 162 .
   Google Scholar
• Diaz , G and Letelier , R . 1993 . Explosive solutions of quasilinear elliptic equations: existence and uniqueness . Nonlinear Anal. , 20 : 97 – 125 .
   Web of Science ®Google Scholar
• Lazer , AC and McKenna , PJ . 1993 . On a problem of Bieberbach and Rademacher . Nonlinear Anal. , 21 : 327 – 335 .
   Web of Science ®Google Scholar
• Lazer , AC and McKenna , PJ . 1996 . On singular boundary value problems for the Monge-Ampere Operator . J. Math. Anal. Appl. , 197 : 341 – 362 .
   Web of Science ®Google Scholar
• Bieberbach , L . 1916 . ▵u = e u und die automorphen Funktionen . Math. Ann. , 77 : 173 – 212 .
   Google Scholar
• Marcus , M and Veron , L . 1993 . Uniqueness of solutions with blow-up at the boundary for a class of nonlinear elliptic equation . C. R. Acad. Sci. Paris , 317 : 559 – 563 .
   Google Scholar
• Pohozaev , SL . 1960 . the Dirichlet problem for the equation ▵u = u 2 . Dokl. Akad. SSSR , 134 : 769 – 772 . (English translation: Sov. Math. 1 (1960), pp. 1143–1146)
   Google Scholar
• Posteraro , MR . 1996 . On the solutions of the equation ▵u = e u blowing up on the boundary . C. R. Acad. Sci. Paris , 322 : 445 – 450 .
   Google Scholar
• Rademacher , H . 1943 . “ Einige besondere problem partieller Differentialgleichungen ” . In Die Differential- und Integralgleichungen, der Mechanik und Physikl , 2nd , Edited by: Frank , P and von Mises , R . 838 – 845 . New York : Rosenberg .
   Google Scholar
• Keller , JB . 1957 . On solutions of ▵u = f(u) . Comm. Pure Appl. Math. , 10 : 503 – 510 .
   Web of Science ®Google Scholar
• Kondrat'ev , VA and Nikishken , VA . 1990 . Asymptotics, near the boundary, of a singular boundary-value problem for a semilinear elliptic equation . Diff. Urav. , 26 : 465 – 468 . (English translation: Differential Equations 26(1990) 345–348)
   Google Scholar
• Loewner , C and Nirenberg , L . 1974 . “ Partial differential equations invariant under conformal or projective transformations ” . In Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers) , Edited by: Ahlfors , LV . 245 – 272 . New York : Academic Press .
   Google Scholar
• Dynkin , EB . 1993 . Superprocesses and partial differential equations . Ann. Probab. , 21 : 1185 – 1262 .
   Web of Science ®Google Scholar
• Dynkin , EB and Kuznetsov , SE . 1996 . Superdiffusions and removable singularities for quasilinear partial differential equations . Comm. Pure Appl. Math. , 49 : 125 – 176 .
   Web of Science ®Google Scholar
• Covei , D-P . Large and entire large solution for a quasilinear problem , Nonlinear Anal, in press
   Google Scholar
• Qishao , L , Zuodong , Y and Twizell , EH . 2004 . Existence of Entire Explosive Positive Solutions of Quasi-linear Elliptic Equations . Appl. Math. Comput. , 148 : 359 – 372 .
   Google Scholar
• Yang , Z . 2006 . Existence of explosive positive solutions of quasilinear elliptic equations . Appl. Math. Comput. , 177 : 581 – 588 .
   Web of Science ®Google Scholar
• Yang , Z , Xu , B and Wu , M . 2007 . Existence of positive boundary blow-up Solutions for Quasilinear Elliptic equations via sub and supersolutions . Appl. Math. Computation , 188 ( 1 ) : 492 – 498 .
   Google Scholar
• Liu , C and Yang , Z . 2007 . Existence of Large Solutions for Quasilinear Elliptic Problems with a Gradient Term . Appl. Math. Comput. , 192 ( 2 ) : 533 – 545 .
   Google Scholar
• Yuan , J and Yang , Z . 2008 . Existence of large solutions for a class of quasilinear elliptic equations . Appl. Math. Comput. , 201 : 852 – 858 .
   Web of Science ®Google Scholar
• Goncalves , JV and Roncalli , A . 2006 . Existence, non-existence and asymptotic behavior of blow-up entire solutions of semilinear elliptic equations . J. Math. Anal. Appl. , 321 : 524 – 536 .
   Google Scholar
• Naito , M . 1984 . A note on bounded positive entire solutions of quasilinear elliptic equations . Hiroshima Math. J. , 14 : 211 – 214 .
   Google Scholar