zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms. (English) Zbl 1189.35104
Summary: We establish results concerning non-existence and existence of entire positive solutions for the nonlinear elliptic problem $$\cases -\Delta_pu= a(x)u^m+\lambda b(x)u^n \quad\text{in }\Bbb R^N,\\ u>0\quad\text{in }\Bbb R^N, \qquad u(x)@>|x|\to\infty>>0, \endcases$$ where $-\infty<m<p-1<n$ with $1<p<N$; $a,b\ge 0$, $a,b\ne 0$ are locally Hölder continuous functions and $\lambda\ge 0$ is a real parameter. The main purpose of this paper, in short, is to complement the principal theorem of {\it B. Xu} and {\it Z. Yang} [Bound. Value Probl. 2007, Article ID 16407, 8 p. (2007; Zbl 1139.35348)] showing existence and non-existence of solutions for the above problem for $\lambda>0$ appropriately.

35J61Semilinear elliptic equations
35J91Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J25Second order elliptic equations, boundary value problems
35J20Second order elliptic equations, variational methods
35J67Boundary values of solutions of elliptic equations
35B09Positive solutions of PDE
Full Text: DOI
[1] Mikljukov, V.: On the asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion, Sbornik mathematics (N.S.) 111 (1980)
[2] Reshetnyak, Y. G.: Index boundedness condition for mappings with bounded distortion, Siberian mathematical journal 9, 281-285 (1968) · Zbl 0167.06602 · doi:10.1007/BF02204791
[3] Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems, Acta Mathematica 138, 219-240 (1977) · Zbl 0372.35030 · doi:10.1007/BF02392316
[4] Herrero, M. A.; Vásquez, J. L.: On the propagation properties of a nonlinear degenerate parabolic equation, Communications in partial differential equations 7, No. 12, 1381-1402 (1982) · Zbl 0516.35041 · doi:10.1080/03605308208820255
[5] Esteban, R.; Vásquez, J. L.: On the equation of turbulent filtration in one-dimensional porous media, Nonlinear analysis 10, No. 11, 1303-1325 (1986) · Zbl 0613.76102 · doi:10.1016/0362-546X(86)90068-4
[6] Serrin, J.; Zou, H.: Cauchy--Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Mathematica 189, 79-142 (2002) · Zbl 1059.35040 · doi:10.1007/BF02392645
[7] Ambrosetti, A.; Brézis, H.; Cerami, G.: Combined effects of concave and convexe nonlinearities in some elliptic problems, Journal of functional analysis 122, 519-543 (1994) · Zbl 0805.35028 · doi:10.1006/jfan.1994.1078
[8] Guedda, M.; Veron, L.: Local and global properties of solutions of quasilinear elliptic equations, Journal of differential equations 76, No. 1, 159-189 (1988) · Zbl 0661.35029 · doi:10.1016/0022-0396(88)90068-X
[9] Guo, Z. M.; Webb, J. R. L.: Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proceedings of the royal society of Edinburgh section A. Mathematics 124, No. 1, 189-198 (1998) · Zbl 0799.35081 · doi:10.1017/S0308210500029280
[10] Bognár, G.; Drábek, P.: The p-Laplacian equation with superlinear and supercritical growth multiplicity of radial solutions, Nonlinear analysis 60, No. 4, 719-728 (2005) · Zbl 1068.34012 · doi:10.1016/j.na.2004.09.047
[11] Prashanth, S.; Sreenadh, K.: Multiplicity of positive solutions for p-Laplace equation with superlinear-type nonlinearities, Nonlinear analysis 56, No. 6 (1998) · Zbl 1100.35038 · doi:10.1016/j.na.2003.10.026
[12] Brezis, H.; Kamin, S.: Sublinear elliptic equations in RN, Manuscripta Mathematica 74, No. 1, 87-106 (1992) · Zbl 0761.35027 · doi:10.1007/BF02567660
[13] Goncalves, J. V.; Santos, C. A.: Existence and asymptotic behavior of non-radially symmetric ground states of semilinear singular elliptic equations, Nonlinear analysis 7, No. 4, 475-490 (2006) · Zbl 1245.35045
[14] Mohammed, A.: Ground state solutions for singular semi-linear elliptic equations, Nonlinear analysis (2008)
[15] Santos, C. A.: On ground state solutions for singular and semi-linear problems including super-linear terms at infinity, Nonlinear analysis 71, No. 12, 6038-6043 (2009) · Zbl 1178.35169 · doi:10.1016/j.na.2009.05.050
[16] Xu, B.; Yang, Z.: Entire bounded solutions for a class of quasilinear elliptic equations, Boundary value problems (2007) · Zbl 1139.35348 · doi:10.1155/2007/16407
[17] Ambrosetti, A.; Brezis, H.; Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems, Journal of functional analysis 122, No. 2, 519-543 (1994) · Zbl 0805.35028 · doi:10.1006/jfan.1994.1078
[18] Bartsch, T.; Willem, M.: On an elliptic equation with concave and convex nonlinearities, Proceedings of the American mathematical society 123, No. 11, 3555-3561 (1995) · Zbl 0848.35039 · doi:10.2307/2161107
[19] Brezis, H.; Oswald, L.: Remarks on sublinear elliptic equations, Nonlinear analysis 10, No. 1, 55-64 (1986) · Zbl 0593.35045 · doi:10.1016/0362-546X(86)90011-8
[20] Goncalves, J. V.; Melo, A. L.; Santos, C. A.: On existence of l\infty-ground states for singular elliptic equations in the presence of a strongly nonlinear term, Advanced nonlinear studies 7, No. 3, 475-490 (2007) · Zbl 1142.35030
[21] Yang, Z.: Existence of positive bounded entire solutions for quasilinear elliptic equations, Applied mathematics and computation 156, No. 3, 743-754 (2004) · Zbl 1108.35336 · doi:10.1016/j.amc.2003.06.024
[22] Ye, D.; Zhou, F.: Invariant criteria for existence of bounded positive solutions, Discrete and continuous dynamical systems, series A 12, No. 3, 413-424 (2005) · Zbl 1080.35028 · doi:10.3934/dcds.2005.12.413
[23] Zhang, Z.: A remark on the existence of positive entire solutions of a sublinear elliptic problem, Nonlinear analysis 67, No. 1, 147-153 (2007) · Zbl 1143.35062 · doi:10.1016/j.na.2006.05.004
[24] Goncalves, J. V.; Santos, C. A.: Positive solutions for a class of quasilinear singular equations, Electronic journal of differential equations 56, 1-15 (2004) · Zbl 1109.35309 · emis:journals/EJDE/Volumes/2004/56/abstr.html
[25] Yang, Z.; Yin, H.: Some news results on the bounded positive entire solutions for quasilinear elliptic equations, Applied mathematics and computation 177, 606-613 (2006) · Zbl 1254.35094
[26] Allegretto, W.; Huang, Y.: A Picone’s identity for the p-Laplacian and applications, Nonlinear analysis 32, No. 7, 819-830 (1998) · Zbl 0930.35053 · doi:10.1016/S0362-546X(97)00530-0
[27] Díaz, J. I.; Saa, J. E.: Existence et unicité de solutions positives pour certains equations elliptiques quasilineaires, C. R. Acad. sci. Paris sér. I math. 305, No. 12, 521-524 (1987) · Zbl 0656.35039