Nonpositone elliptic problems in

Authors:
W. Allegretto and P. O. Odiobala

Journal:
Proc. Amer. Math. Soc. **123** (1995), 533-541

MSC:
Primary 35J65; Secondary 35B50

DOI:
https://doi.org/10.1090/S0002-9939-1995-1219715-9

MathSciNet review:
1219715

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of the existence of positive decaying solutions in to the nonlinear equation where *l* denotes a second-order uniformly elliptic operator and is superlinear and subcritical with . Existence is obtained by Mountain Pass Arguments and positivity by establishing bounds for *u* in various Sobolev norms and by comparison with the case .

**[1]**W. Allegretto, P. Nistri, and P. Zecca,*Positive solutions of elliptic non-positone problems*, Differential Integral Equations**5**(1992), 95-101. MR**1141729 (92k:35088)****[2]**W. Allegretto and L. S. Yu,*Positive*-*solutions of subcritical nonlinear problems*, J. Differential Equations**87**(1990), 340-352. MR**1072905 (91k:35014)****[3]**A. Bahri and J. M. Coron,*The scalar-curvature problem on the standard three-dimensional sphere*, J. Functional Anal.**95**(1991), 106-172. MR**1087949 (92k:58055)****[4]**A. Bahri and P. L. Lions,*Solutions of superlinear elliptic equations and their Morse indices*, Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, Pitman Res. Notes Math. Ser., vol. 243, Longman Sci. Tech., Harlow, 1992. MR**1176341 (93g:35044)****[5]**K. J. Brown, A. Castro, and R. Shivaji,*Non-existence of radially symmetric non-negative solutions for a class of semi-positone problems*, Differential Integral Equations**2**(1989), 541-545. MR**996760 (90f:35010)****[6]**K. J. Brown and R. Shivaji,*Instability of nonnegative solutions for a class of semipositone problems*, Proc. Amer. Math. Soc.**112**(1991), 121-124. MR**1043405 (91h:35047)****[7]**A. Castro and R. Shivaji,*Nonnegative solutions for a class of nonpositone problems*, Proc. Roy. Soc. Edinburgh Sect. A**108**(1988), 291-302. MR**943804 (90m:34040)****[8]**-,*Nonnegative solutions for a class of radially symmetric nonpositone problems*, Proc. Amer. Math. Soc.**106**(1989), 735-740. MR**949875 (89k:35015)****[9]**V. Coti Zelati and P. H. Rabinowitz,*Homoclinic type solutions for a semilinear elliptic PDE on*, Comm. Pure Appl. Math.**45**(1993), 1217-1269. MR**1181725 (93k:35087)****[10]**D. Gilbarg and N. S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Springer-Verlag, Berlin, Heidleberg, New York, and Tokyo, 1983. MR**737190 (86c:35035)****[11]**Y. Li and W.-M. Ni,*On the asymptotic behaviour and radial symmetry of positive solutions of semilinear elliptic equations in*, Arch. Rational Mech. Anal.**118**(1992), 195-222. MR**1158935 (93e:35036)****[12]**Y. Y. Li,*On**in*, Comm. Pure Appl. Math.**45**(1993), 303-340.**[13]**P. L. Lions,*On positive solutions of semilinear elliptic equations in unbounded domains*, Nonlinear Diffusion Equations and Their Equilibrium States (W.-M. Ni, L. A. Peletier, and J. Serrin, eds.), Springer-Verlag, New York, 1988, pp. 85-122. MR**956083 (89m:35021)****[14]**W. Littman, G. Stampacchia, and H. F. Weinberger,*Regular points for elliptic equations with discontinuous coefficients*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**17**(1963), 43-77. MR**0161019 (28:4228)****[15]**E. S. Noussair and C. A. Swanson,*An*-*theory of subcritical semilinear elliptic problems*, J. Differential Equations**84**(1990), 52-61. MR**1042658 (91d:35074)****[16]**P. H. Rabinowitz,*Minimax methods in critical point theory with applications to differential equations*, Amer. Math. Soc., Providence, RI, 1986. MR**845785 (87j:58024)****[17]**-,*A note on a semilinear elliptic equation on*, Nonlinear Analysis, a Tribute in Honour of Giovanni Prodi (A. Ambrosetti and A. Marino, eds.), Quaderni, Scuola Normale Superiore, Pisa, 1991, pp. 307-318.**[18]**W. Rother,*Nonlinear scalar field equations*, Differential Integral Equations**5**(1992), 747-767. MR**1167494 (93e:35030)****[19]**J. Smoller and A. Wasserman,*Symmetry-breaking for semilinear elliptic equations with general boundary conditions*, Comm. Math. Phys.**105**(1986), 415-441. MR**848648 (87j:35048)****[20]**-,*Symmetry-breaking for positive solutions of semilinear elliptic equations*, Arch. Rational Mech. Anal.**95**(1986), 217-225. MR**853965 (88e:35080b)****[21]**-,*Existence of positive solutions for semilinear elliptic equations in general domains*, Arch. Rational. Mech. Anal.**98**(1987), 229-249. MR**867725 (88a:35101)****[22]**G. Stampacchia,*Equations elliptiques du second order à coefficients discontinues*, Presses Univ. Montréal, Montréal, 1966. MR**0251373 (40:4603)****[23]**S. Unsurangsie,*Existence of a solution for a wave equation and elliptic problems*, Ph.D. Thesis (A. Castro, supervisor), University of North Texas, 1988.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
35J65,
35B50

Retrieve articles in all journals with MSC: 35J65, 35B50

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1219715-9

Keywords:
Nonpositone,
elliptic,
positive decaying solution

Article copyright:
© Copyright 1995
American Mathematical Society