## Nonpositone elliptic problems in $\textbf {R}^ n$

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- by W. Allegretto and P. O. Odiobala PDF
- Proc. Amer. Math. Soc.
**123**(1995), 533-541 Request permission

## 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

- W. Allegretto, P. Nistri, and P. Zecca,
*Positive solutions of elliptic nonpositone problems*, Differential Integral Equations**5**(1992), no. 1, 95–101. MR**1141729** - W. Allegretto and L. S. Yu,
*Positive $L^p$-solutions of subcritical nonlinear problems*, J. Differential Equations**87**(1990), no. 2, 340–352. MR**1072905**, DOI 10.1016/0022-0396(90)90006-B - A. Bahri and J.-M. Coron,
*The scalar-curvature problem on the standard three-dimensional sphere*, J. Funct. Anal.**95**(1991), no. 1, 106–172. MR**1087949**, DOI 10.1016/0022-1236(91)90026-2 - 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 (L’Aquila, 1990) Pitman Res. Notes Math. Ser., vol. 243, Longman Sci. Tech., Harlow, 1992, pp. 10–20. MR**1176341** - K. J. Brown, Alfonso Castro, and R. Shivaji,
*Nonexistence of radially symmetric nonnegative solutions for a class of semi-positone problems*, Differential Integral Equations**2**(1989), no. 4, 541–545. MR**996760** - K. J. Brown and R. Shivaji,
*Instability of nonnegative solutions for a class of semipositone problems*, Proc. Amer. Math. Soc.**112**(1991), no. 1, 121–124. MR**1043405**, DOI 10.1090/S0002-9939-1991-1043405-5 - Alfonso Castro and R. Shivaji,
*Nonnegative solutions for a class of nonpositone problems*, Proc. Roy. Soc. Edinburgh Sect. A**108**(1988), no. 3-4, 291–302. MR**943804**, DOI 10.1017/S0308210500014670 - Alfonso Castro and R. Shivaji,
*Nonnegative solutions for a class of radially symmetric nonpositone problems*, Proc. Amer. Math. Soc.**106**(1989), no. 3, 735–740. MR**949875**, DOI 10.1090/S0002-9939-1989-0949875-3 - Vittorio Coti Zelati and Paul H. Rabinowitz,
*Homoclinic type solutions for a semilinear elliptic PDE on $\textbf {R}^n$*, Comm. Pure Appl. Math.**45**(1992), no. 10, 1217–1269. MR**1181725**, DOI 10.1002/cpa.3160451002 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - Yi Li and Wei-Ming Ni,
*On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $\textbf {R}^n$. I. Asymptotic behavior*, Arch. Rational Mech. Anal.**118**(1992), no. 3, 195–222. MR**1158935**, DOI 10.1007/BF00387895
Y. Y. Li, - P.-L. Lions,
*On positive solutions of semilinear elliptic equations in unbounded domains*, Nonlinear diffusion equations and their equilibrium states, II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 13, Springer, New York, 1988, pp. 85–122. MR**956083**, DOI 10.1007/978-1-4613-9608-6_{6} - W. Littman, G. Stampacchia, and H. F. Weinberger,
*Regular points for elliptic equations with discontinuous coefficients*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**17**(1963), 43–77. MR**161019** - Ezzat S. Noussair and Charles A. Swanson,
*An $L^q(\textbf {R}^N)$-theory of subcritical semilinear elliptic problems*, J. Differential Equations**84**(1990), no. 1, 52–61. MR**1042658**, DOI 10.1016/0022-0396(90)90126-A - Paul H. Rabinowitz,
*Minimax methods in critical point theory with applications to differential equations*, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**845785**, DOI 10.1090/cbms/065
—, - W. Rother,
*Nonlinear scalar field equations*, Differential Integral Equations**5**(1992), no. 4, 777–792. MR**1167494** - Joel A. Smoller and Arthur G. Wasserman,
*Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions*, Comm. Math. Phys.**105**(1986), no. 3, 415–441. MR**848648** - Joel Smoller and Arthur Wasserman,
*An existence theorem for positive solutions of semilinear elliptic equations*, Arch. Rational Mech. Anal.**95**(1986), no. 3, 211–216. MR**853964**, DOI 10.1007/BF00251358 - Joel Smoller and Arthur Wasserman,
*Existence of positive solutions for semilinear elliptic equations in general domains*, Arch. Rational Mech. Anal.**98**(1987), no. 3, 229–249. MR**867725**, DOI 10.1007/BF00251173 - Guido Stampacchia,
*Èquations elliptiques du second ordre à coefficients discontinus*, Séminaire de Mathématiques Supérieures, No. 16 (Été, vol. 1965, Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR**0251373**
S. Unsurangsie,

*On*$- \Delta u = k(x){u^5}$

*in*${\mathbb {R}^3}$, Comm. Pure Appl. Math.

**45**(1993), 303-340.

*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.

*Existence of a solution for a wave equation and elliptic problems*, Ph.D. Thesis (A. Castro, supervisor), University of North Texas, 1988.

## Additional Information

- © Copyright 1995 American Mathematical Society
- 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