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Exact multiplicity of positive solutions for a class of semilinear problems. (English) Zbl 0918.35049
By carefully analyzing the local behaviour of the positive solution curve at turning points, the authors precisely characterized the global bifurcation diagrams of an ODE arising from the radial positive solutions for a class of semilinear elliptic problems, and then established an exact multiplicity result of positive solutions in a ball. The prototype examples of the nonlinear terms are: \[ f(u)= u(u- b)(c-u),\tag{1} \] where \(0< 2b< c\), \[ f(u)= u^p- u^q,\tag{2} \] where \(0< p< q\), and \[ f(u)= u(u- b)/(1+ au^p),\tag{3} \] where \(a> 0\), \(b\geq 0\), and \(1< p\leq 2\). Many of these bifurcation diagrams were formally discussed by P. L. Lions [SIAM Rev. 24, 441-467 (1982; Zbl 0511.35033)], but proved rigorously for special nonlinearities in balls in this paper.

MSC:
35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
35J20 Variational methods for second-order elliptic equations
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[1] Ali, I.; Castro, A.; Shivaji, R., Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. amer. math. soc., 117, 775-781, (1993) · Zbl 0770.34019
[2] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations 1, 2, Arch. rational mech. anal., 82, 313-375, (1983)
[3] Beresytcki, H.; Lions, P.L.; Peletier, L.A., An ODE approach to the existence of positive solutions for semilinear problem inR^{n}, Indiana univ. math. J., 30, 141-167, (1981)
[4] Castro, A.; Gadam, S., Uniqueness of stable and unstable positive solutions for semipositone problems, Nonlinear anal., 22, 425-429, (1994) · Zbl 0804.35038
[5] Crandall, M.G.; Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. rational mech. anal., 52, 161-180, (1973) · Zbl 0275.47044
[6] Dancer, E.N., On positive solutions of some singular perturbed problems where the nonlinearity changes sign, Topological methods nonlinear anal., 5, 141-175, (1995) · Zbl 0835.35013
[7] E. N. Dancer, A note on asymptotic uniqueness for some nonlinearities which change sign, 1995
[8] Gardner, R.; Peletier, L.A., The set of positive solutions of semilinear equations in large balls, Proc. royal. soc. Edinburgh, 104A, 53-72, (1986) · Zbl 0625.35030
[9] Gidas, B.; Ni, W.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[10] Hale, J., Asymptotic behavior of dissipative systems, Math. surveys, 25, (1988) · Zbl 0642.58013
[11] Holzmann, M.; Kielhöfer, H., Uniqueness of global positive solution branches of nonlinear elliptic problem, Math. ann., 300, 221-241, (1994) · Zbl 0806.35034
[12] Korman, P.; Li, Y.; Ouyang, T., Exact multiplicity results for boundary-value problems with nonlinearities generalizing cubic, Proc. roy. soc. Edinburgh sect. A, 126, 599-616, (1996) · Zbl 0855.34022
[13] Korman, P.; Li, Y.; Ouyang, T., An exact multiplicity result for a class of semilinear equations, Comm. partial differential equations, 22, 661-684, (1997) · Zbl 0877.35048
[14] Korman, P.; Ouyang, T., Exact multiplicity results for two classes of boundary-value problems, Differential integral equations, 6, 1507-1517, (1993) · Zbl 0780.34013
[15] Korman, P.; Ouyang, T., Multiplicity results for two classes of boundary-value problems, SIAM J. math. analysis, 26, 180-189, (1995) · Zbl 0824.34028
[16] Kwong, M., Uniqueness of positive solutions ofδuuup=0 in{\bfr}n, Arch. rational mech. anal., 105, 243-266, (1989)
[17] Kwong, M.; Zhang, L., Uniqueness of the positive solution ofδufu, Differential integral equations, 4, 583-599, (1991) · Zbl 0724.34023
[18] Lions, P.L., On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 441-467, (1982) · Zbl 0511.35033
[19] Lin, C.; Ni, W., A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. amer. math. soc., 102, 271-277, (1988) · Zbl 0652.35085
[20] Mcleod, K.; Serrin, J., Uniqueness of the positive radial solutions ofδufurn, Arch. rational mech. anal., 99, 115-145, (1987) · Zbl 0667.35023
[21] Ni, W.; Nussbaum, R.D., Uniqueness and nonuniqueness for positive radial solutions ofδufur, Comm. pure appl. math., 38, 67-108, (1985) · Zbl 0581.35021
[22] T. Ouyang, J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem: II, 1996 · Zbl 0947.35067
[23] Peletier, L.A.; Serrin, J., Uniqueness of positive solutions of semilinear equations inR^{n}, Arch. rational mech. anal., 81, 181-197, (1983) · Zbl 0516.35031
[24] Peletier, L.A.; Serrin, J., Uniqueness of solutions of semilinear equations inR^{n}, J. differential equations, 61, 380-397, (1986) · Zbl 0577.35035
[25] Rabinowitz, P.H., On bifurcation from infinity, J. differential equations, 14, 462-475, (1973) · Zbl 0272.35017
[26] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS reg. conf. ser. in math., (1986) · Zbl 0609.58002
[27] Schaaf, R.; Schmitt, K., Asymptotic behavior of positive branches of elliptic problems with linear part at resonance, Z. angew. math. phys., 43, 645-676, (1992) · Zbl 0759.35011
[28] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. differential equations, 39, 269-290, (1981) · Zbl 0425.34028
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