Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Nonpositone elliptic problems in $ {\bf R}\sp n$

Authors: W. Allegretto and P. O. Odiobala
Journal: Proc. Amer. Math. Soc. 123 (1995), 533-541
MSC: Primary 35J65; Secondary 35B50
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 $ {\mathbb{R}^n}$ to the nonlinear equation $ lu = \lambda f(x,u)$ where l denotes a second-order uniformly elliptic operator and $ f(x,u)$ is superlinear and subcritical with $ f(x,0) \leq 0$. Existence is obtained by Mountain Pass Arguments and positivity by establishing bounds for u in various Sobolev norms and by comparison with the case $ l = - \Delta $.

References [Enhancements On Off] (What's this?)

  • [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 $ {L^p}$-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 $ {\mathbb{R}^n}$, 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 $ {\mathbb{R}^n}$, Arch. Rational Mech. Anal. 118 (1992), 195-222. MR 1158935 (93e:35036)
  • [12] Y. Y. Li, On $ - \Delta u = k(x){u^5}$ in $ {\mathbb{R}^3}$, 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 $ {L^q}({\mathbb{R}^n})$-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 $ {\mathbb{R}^n}$, 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.

Similar Articles

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

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

Additional Information

Keywords: Nonpositone, elliptic, positive decaying solution
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society