Existence and multiplicity results for a semilinear elliptic eigenvalue problem

Clément, Philippe; Sweers, Guido

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 14 (1987) no. 1, pp. 97-121.

[1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), pp. 620-709. | MR | Zbl

[2] A. Ambrosetti, P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), pp. 411-422. | MR | Zbl

[3] S.B. Angenent, Uniqueness of the solution of a semilinear boundary value problem, Math. Ann. 272 (1985), pp. 129-138. | MR | Zbl

[4] H. Bérestycki, P.-L. Lions, Une méthode locale pour l'existence de solutions positives de problèmes semilinéaires elliptiques dans R N, J. Analyse Math. 38 (1980), pp. 144-187. | MR | Zbl

[5] P. Clément, L.A. Peletier, Positive superharmonic solutions to semilinear elliptic eigenvalue problems, J. Math. Anal. Appl. 100 (1984), pp. 561-582. | MR | Zbl

[6] P. Clément, G. Sweers, Existence et multiplicité des solutions d'un problème aux valeurs propres elliptiques semilinéaire, C.R. Acad Sci. Paris 302, Série 1, 19 (1986), pp. 681-683. | MR | Zbl

[7] E.N. Dancer, Uniqueness for elliptic equations when a parameter is large, Nonlinear Anal. Theory, Methods and Appl. 8 (1984), pp. 835-836. | Zbl

[8] E.N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, to appear in Proc. London Math. Soc. | MR | Zbl

[9] E.N. Dancer, Multiple fixed points of positive mappings, Preprint (1985). | MR

[10] D. De Figueiredo, On the uniqueness of positive solutions of the Dirichlet problem -Δu = λ sin u, Nonlinear P.D.E. and Appl. Collège de France Seminar Vol. 7, Pitman (1985), pp. 80-83. | Zbl

[11] R. Gardner, L.A. Peletier, The set of positive solutions of semilinear equations in large balls, to appear. | MR | Zbl

[12] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), pp. 209-243. | MR | Zbl

[13] P. Hess, On multiple solutions of nonlinear. elliptic eigenvalue problems, Comm. Partial Differential Equations 6 (1981), pp. 951-961. | MR | Zbl

[14] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), pp. 441-467. | MR | Zbl

[15] W.M. Ni, J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case, Univ. of Minnesota Math. Rep. (1984), pp. 84-150.

[16] L. Nirenberg, Topics in nonlinear functional analysis, Lecture Notes Courant Inst. of Math. Sci, New York Univ. (1974). | MR | Zbl

[17] S.I. Pohozaev, Eigenfunctions of the equations Δu + λf (u) = 0, Soviet Math. Dokl., 6 (1965), pp. 1408-1411. | Zbl

[18] M.H. Protter, H.F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ (1967). | MR | Zbl

[19] P.H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J. 23 (1973), pp. 172-185. | MR | Zbl

[20] P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), pp. 161-202. | MR | Zbl

[21] D.H. Sattinger, Topics in stability and bifurcation theory, Lect. Notes in Math. 309, Springer, Berlin Heidelberg New York (1973). | MR | Zbl

[22] J. Serrin, Nonlinear equations of second order, A.M.S. Sympos. Partial Differential Equations, Berkeley, August 1971.

[23] F. Trèves, Basic linear partial differential equations, Academic Press, New York (1975). | MR | Zbl

[24] E.N. Dancer, K. Schmitt, On positive solutions of semilinear elliptic equations, Preprint (1986). | MR