Existence of decaying entire solutions of a class of semilinear elliptic equations
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- by Takaŝi Kusano, Ezzat S. Noussair and Charles A. Swanson PDF
- Proc. Amer. Math. Soc. 104 (1988), 1141-1147 Request permission
Abstract:
The main result establishes the existence of a nontrivial nonnegative radial solution $u \in C_{\operatorname {loc} }^2({{\mathbf {R}}^N})$ of a semilinear elliptic eigenvalue problem in ${{\mathbf {R}}^N},N \geq 3$, such that $u(|x|)$ has uniform limit zero as $|x| \to \infty$. Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space $W_0^{1,2}({{\mathbf {R}}^N})$, a considerable departure from standard procedures is required.References
- H. Berestycki and P.-L. Lions, Une méthode locale pour l’existence de solutions positives de problèmes semi-linéaires elliptiques dans $\textbf {R}^{N}$, J. Analyse Math. 38 (1980), 144–187 (French). MR 600785, DOI 10.1007/BF03033880
- H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI 10.1007/BF00250555
- H. Berestycki, P.-L. Lions, and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\textbf {R}^{N}$, Indiana Univ. Math. J. 30 (1981), no. 1, 141–157. MR 600039, DOI 10.1512/iumj.1981.30.30012
- Melvyn S. Berger, On the existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. Functional Analysis 9 (1972), 249–261. MR 0299966, DOI 10.1016/0022-1236(72)90001-8
- Wei Yue Ding and Wei-Ming Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), no. 4, 283–308. MR 807816, DOI 10.1007/BF00282336
- B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
- Nichiro Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), no. 1, 125–158. MR 750393
- Takaŝi Kusano and Manabu Naito, Positive entire solutions of superlinear elliptic equations, Hiroshima Math. J. 16 (1986), no. 2, 361–366. MR 855164
- Takaŝi Kusano and Shinnosuke Oharu, On entire solutions of second order semilinear elliptic equations, J. Math. Anal. Appl. 113 (1986), no. 1, 123–135. MR 826663, DOI 10.1016/0022-247X(86)90337-9
- Wei Ming Ni, On the elliptic equation $\Delta u+K(x)u^{(n+2)/(n-2)}=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529. MR 662915, DOI 10.1512/iumj.1982.31.31040
- E. S. Noussair and C. A. Swanson, Oscillation theory for semilinear Schrödinger equations and inequalities, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975/76), no. 1, 67–81. MR 442436, DOI 10.1017/S0080454100012541 —, Positive solutions of semilinear elliptic problems in unbounded domains, J. Differential Equations 57 (1985), 349-372.
- Ezzat S. Noussair and Charles A. Swanson, Positive solutions of elliptic systems with bounded nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 3-4, 321–332. MR 943806, DOI 10.1017/S0308210500014694
- Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 454365
- John F. Toland, On positive solutions of $-\Delta u=F(x,\,u)$, Math. Z. 182 (1983), no. 3, 351–357. MR 696532, DOI 10.1007/BF01179755
- J. F. Toland, Positive solutions of nonlinear elliptic equations—existence and nonexistence of solutions with radial symmetry in $L_{p}(\textbf {R}^{N})$, Trans. Amer. Math. Soc. 282 (1984), no. 1, 335–354. MR 728716, DOI 10.1090/S0002-9947-1984-0728716-3
- James S. W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. 17 (1975), 339–360. MR 367368, DOI 10.1137/1017036
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 1141-1147
- MSC: Primary 35J60; Secondary 35B40
- DOI: https://doi.org/10.1090/S0002-9939-1988-0929405-1
- MathSciNet review: 929405