Existence and multiplicity results for a semilinear elliptic eigenvalue problem

 Access to full text

 Full (PDF)

1. [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev.18 (1976), pp. 620-709. Zbl0345.47044MR415432
2. [2] A. Ambrosetti, P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl.73 (1980), pp. 411-422. Zbl0433.35026MR563992
3. [3] S.B. Angenent, Uniqueness of the solution of a semilinear boundary value problem, Math. Ann.272 (1985), pp. 129-138. Zbl0576.35044MR794096
4. [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. Zbl0518.35034MR600785
5. [5] P. Clément, L.A. Peletier, Positive superharmonic solutions to semilinear elliptic eigenvalue problems, J. Math. Anal. Appl.100 (1984), pp. 561-582. Zbl0543.35038MR743342
6. [6] P. Clément, G. Sweers, Existence et multiplicité des solutions d'un problème aux valeurs propres elliptiques semilinéaire, C.R. Acad Sci. Paris302, Série 1, 19 (1986), pp. 681-683. Zbl0606.35035MR847753
7. [7] E.N. Dancer, Uniqueness for elliptic equations when a parameter is large, Nonlinear Anal. Theory, Methods and Appl.8 (1984), pp. 835-836. Zbl0526.35032
8. [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. Zbl0572.35040MR868453
9. [9] E.N. Dancer, Multiple fixed points of positive mappings, Preprint (1985). Zbl0597.47034MR859319
10. [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. Zbl0569.35037
11. [11] R. Gardner, L.A. Peletier, The set of positive solutions of semilinear equations in large balls, to appear. Zbl0625.35030MR877892
12. [12] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys.68 (1979), pp. 209-243. Zbl0425.35020MR544879
13. [13] P. Hess, On multiple solutions of nonlinear. elliptic eigenvalue problems, Comm. Partial Differential Equations6 (1981), pp. 951-961. Zbl0468.35073MR619753
14. [14] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev.24 (1982), pp. 441-467. Zbl0511.35033MR678562
15. [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. [16] L. Nirenberg, Topics in nonlinear functional analysis, Lecture Notes Courant Inst. of Math. Sci, New York Univ. (1974). Zbl0286.47037MR488102
17. [17] S.I. Pohozaev, Eigenfunctions of the equations Δu + λf (u) = 0, Soviet Math. Dokl., 6 (1965), pp. 1408-1411. Zbl0141.30202
18. [18] M.H. Protter, H.F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ (1967). Zbl0153.13602MR219861
19. [19] P.H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J.23 (1973), pp. 172-185. Zbl0264.35032MR318667
20. [20] P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math.3 (1973), pp. 161-202. Zbl0255.47069MR320850
21. [21] D.H. Sattinger, Topics in stability and bifurcation theory, Lect. Notes in Math. 309, Springer, BerlinHeidelbergNew York (1973). Zbl0248.35003MR463624
22. [22] J. Serrin, Nonlinear equations of second order, A.M.S. Sympos. Partial Differential Equations, Berkeley, August 1971. Zbl0271.35004
23. [23] F. Trèves, Basic linear partial differential equations, Academic Press, New York (1975). Zbl0305.35001MR447753
24. [24] E.N. Dancer, K. Schmitt, On positive solutions of semilinear elliptic equations, Preprint (1986). Zbl0661.35031MR908646

1. E. N. Dancer, Shusen Yan, A minimization problem associated with elliptic systems of Fitz–Hugh–Nagumo type
2. Henri Berestycki, Luis Caffarelli, Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains
3. Edward Norman Dancer, Shusen Yan, Peak solutions for an elliptic system of FitzHugh-Nagumo type