Nonpositone elliptic problems in 𝑅ⁿ

Allegretto, W.; Odiobala, P. O.

doi:10.1090/S0002-9939-1995-1219715-9

Nonpositone elliptic problems in $\textbf {R}^ n$
HTML articles powered by AMS MathViewer

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

On

in

45

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

A note on a semilinear elliptic equation on

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

Existence of a solution for a wave equation and elliptic problems