Ding, Wei-Yue; Ni, Wei-Ming On the existence of positive entire solutions of a semilinear elliptic equation. (English) Zbl 0616.35029 Arch. Ration. Mech. Anal. 91, 283-308 (1986). Existence of positive entire solutions to the equation \(\Delta u- u+Q(x)u^ p=0\) in \({\mathbb{R}}^ n\) is shown, where \(p>1\) and Q(x) is a given potential, which, in general, is not radially symmetric. The approach is by establishing existence of positive solutions in any ball with radius R centered at the origin and then passing to the limit \(R\to \infty\). Reviewer: R.Kreß Cited in 4 ReviewsCited in 211 Documents MSC: 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:semilinear elliptic equation; Existence; positive entire solutions; potential × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Ambrosetti, A., & P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 [2] Berestycki, H., & P.-L. Lions, Nonlinear scalar field equations I, II, Arch. Rational Mech. Anal. 82 (1983), 313–345, 347–375. · Zbl 0533.35029 [3] Berger, M., On the existence and structure of stationary states fora nonlinear Klein-Gordon equation, J. Funct. Anal. 9 (1972), 249–261. · Zbl 0224.35061 · doi:10.1016/0022-1236(72)90001-8 [4] Ding, W.-Y., & W.-M. NI, On the elliptic equation \(\Delta u{\text{ + }}K{\text{ }}u^{\frac{{n + 2}}{{n - 2}}} {\text{ = 0}}\) and related topics, Duke Math. J. 52 (1985), 485–506. · Zbl 0592.35048 · doi:10.1215/S0012-7094-85-05224-X [5] Gidas, B., W.-M. NI, & L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbb{R}\)n, Advances in Math. Supplementary Studies 7A (1981), 369–402. · Zbl 0469.35052 [6] Gidas, B., W.-M. Ni, & L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243. · Zbl 0425.35020 · doi:10.1007/BF01221125 [7] Kato, T., Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425. · Zbl 0091.09502 · doi:10.1002/cpa.3160120302 [8] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, II. Annales de l’Institut Henri Poincaré – Analyse non linéaire 1 (1984), 223–283. · Zbl 0704.49004 [9] Nehari, Z., On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish Acad. 62 (1963), 117–135. · Zbl 0124.30204 [10] Ni, W.-M., On the elliptic equation \(\Delta u{\text{ + }}K{\text{(}}x{\text{) }}u^{\frac{{n + 2}}{{n - 2}}} {\text{ = 0}}\) , its generalizations and applications to geometry, Indiana Univ. Math. J. 31 (1982), 493–529. · Zbl 0496.35036 · doi:10.1512/iumj.1982.31.31040 [11] Rabinowitz, P., Variational methods for nonlinear eigenvalue problems, Eigenvalues of Nonlinear Problems, C.I.M.E. Edizioni Cremonese 1974. · Zbl 0278.35040 [12] Serrin, J., A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. · Zbl 0222.31007 · doi:10.1007/BF00250468 [13] Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162. · Zbl 0356.35028 · doi:10.1007/BF01626517 [14] Stuart, C. A., Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. (3) 45 (1982), 169–192. · Zbl 0505.35010 · doi:10.1112/plms/s3-45.1.169 [15] Zhang, D., Private communication. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.