## Multiple solutions for a class of nonlinear Sturm-Liouville problems on the half line.(English)Zbl 0703.34032

The author considers a class of nonlinear Sturm-Liouville problems. He gives an extensive list of references containing fourty references together with the complete proofs of the following: 7 theorems, 17 remarks, 17 lemmas, 5 propositions and two corollaries which are pieced together to construct essentially the author’s doctoral thesis from the University of Wisconsin Madison (USA) where the reviewer worked as a visiting professor for fourteen months. In the first section the author obtains existence and uniqueness for positive and negative solutions for the problems $(1)\quad -u^ n=\lambda r(x)u-F(x,u)u,\quad 0<x<+\infty,\quad u(a)\cos \theta -u'(a) \sin \theta =0,\quad u\in L^ 2[a,\infty),$ where r and F are nonnegative continuous functions, $$a\geq 0$$ and $$0\leq \theta \leq \pi /2.$$
In section 2 he obtains the existence of solutions with prescribed number of nodes when the nonlinearitiy is odd near zero. In section 3 the author considers analogous results for radical solutions in higher dimensional class, while in section 4 he considers the partial differential equation $(2)\quad \Delta \hat u=\lambda \hat r(x)\hat u(x)-\hat F(x,\hat u(x))\hat u(x),\quad x\in {\mathbb{R}},$ where $$\hat r:$$ $${\mathbb{R}}^ N\to (0,\infty)$$ and $$\hat F:$$ $${\mathbb{R}}^ N\times {\mathbb{R}}\to (0,\infty)$$ are radially symmetric. The author seeks the solution of (2) in the form $$\hat u\in L^ 2({\mathbb{R}}^ N)\cap C^ 2({\mathbb{R}}^ N)$$.
Reviewer: F.M.Ragab

### MSC:

 34B24 Sturm-Liouville theory 34A34 Nonlinear ordinary differential equations and systems

### Keywords:

nonlinear Sturm-Liouville problems
Full Text:

### References:

  Amick, C.J, Semilinear elliptic eigenvalue problems on an infinite strip with an application to stratified fluids, Ann. scuola norm. sup. Pisa cl. sci. (4), 11, 441-499, (1984) · Zbl 0568.35076  Benci, V; Fortunato, D, Does bifurcation from the essential spectrum occur?, Comm. partial differential equations, 6, 249-272, (1981) · Zbl 0471.35007  Berestycki, H; Lions, P.-L, Nonlinear scalar field equations, Arch. rational mech. anal., 82, 313-375, (1983)  Birkhoff, G; Rota, G, Ordinary differential equations, (1978), Wiley New York · Zbl 0183.35601  Bongers, A; Heinz, H.-P; Küpper, T, Existence and bifurcation theorems for non-linear elliptic eigenvalue problems on unbounded domains, J. differential equations, 47, 327-357, (1983) · Zbl 0506.35081  Chen, C.N, Multiple solutions and bifurcation for a class of nonlinear Sturm-Liouville eigenvalue problems on an unbounded domain, ()  {\scC. N. Chen}, Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems, Arch. Rational Mech. Anal., in press. · Zbl 0712.34035  {\scC. N. Chen}, Multiple solutions for a class of nonlinear Sturm-Liouville problems when nonlinearities are not odd, J. Differential Equations, in press. · Zbl 0779.34018  Chiapinelli, R; Stuart, C.A, Bifurcation when the linearized problem has no eigenvalues, J. differential equations, 30, 296-307, (1978) · Zbl 0419.34010  Coddington, E.A; Levinson, N, Theory of ordinary differential equations, (1955), McGraw-Hill New York · Zbl 0042.32602  Courant, R; Hilbert, D, ()  Crandall, M.G; Rabinowitz, P.H, Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. math. mech., 29, 1083-1102, (1970) · Zbl 0206.09705  Esteban, M.J, Multiple solutions of semilinear elliptic problems in a ball, J. differential equations, 57, 112-137, (1985) · Zbl 0519.35031  Gilbarg, D; Trudinger, N.S, Elliptic partial differential equations of second order, (1977), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0691.35001  Hartman, P, The L^{2}-solution of linear differential equations of second order, Duke math. J., 14, 323-326, (1947) · Zbl 0029.29401  Heinz, H.-P, Nodal properties and variational characterizations of solutions to nonlinear Sturm-Liouville problems, J. differential equations, 62, 299-333, (1986) · Zbl 0579.34011  Heinz, H.-P, Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems, J. differential equations, 64, 79-108, (1986) · Zbl 0593.34021  Heinz, H.-P, Free Ljusternik-schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems, J. differential equations, 66, 263-300, (1987) · Zbl 0607.34012  Hempel, J.A, Multiple solutions for a class of nonlinear boundary value problems, Indiana univ. math. J., 20, 983-996, (1971) · Zbl 0225.35045  Hempel, J.A, Superlinear variational boundary value problems and nonuniqueness, ()  Jones, C; Küpper, T, Characterization of bifurcation from the continuous spectrum by nodal properties, J. differential equations, 54, 196-220, (1984) · Zbl 0547.34018  Jones, C; Küpper, T, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. math. anal., 17, No. 4, 803-835, (1986) · Zbl 0606.35032  Küpper, T, The lowest point of the continuous spectrum as a bifurcation point, J. differential equations, 34, 212-217, (1979) · Zbl 0388.47040  Küpper, T, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Math. methods appl. sci., 1, 572-580, (1979) · Zbl 0437.34010  Küpper, T; Riemer, D, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonlinear anal. (TMA), 3, 555-561, (1979) · Zbl 0388.47039  Nehari, Z, Characteristic values associated with a class of nonlinear second-order differential equations, Acta math., 105, 141-175, (1961) · Zbl 0099.29104  Protter, M.H; Weinberger, H.F, Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602  Rabinowitz, P.H, A note on a nonlinear eigenvalue problem for a class of differential equations, J. differential equations, 9, 536-548, (1971) · Zbl 0218.34027  Rabinowitz, P.H, Global aspects of bifurcation, () · Zbl 0212.16504  Rabinowitz, P.H, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. pure appl. math., 23, 939-961, (1970) · Zbl 0206.09706  Rabinowitz, P.H, Some global results for nonlinear eigenvalue problems, J. fund. anal., 7, 487-513, (1971) · Zbl 0212.16504  Rabinowitz, P.H, Some aspects of nonlinear eigenvalue problems, Rocky mountain J. math., 3, 161-202, (1973) · Zbl 0255.47069  Rudin, W, Principles of mathematical analysis, (1976), McGraw-Hill New York · Zbl 0148.02903  Ryder, G.H, Boundary value problems for a class of nonlinear differential equations, Pacific J. math., 22, 477-503, (1967) · Zbl 0152.28303  Stuart, C.A, Bifurcation from the essential spectrum, () · Zbl 0888.47045  Stuart, C.A, Bifurcation for Dirichlet problems without eigenvalues, (), 169-192 · Zbl 0505.35010  Stuart, C.A, Bifurcation for Neumann problems without eigenvalues, J. differential equations, 36, 391-407, (1980) · Zbl 0468.34009  Stuart, C.A, Global properties of components of solutions of non-linear second order ordinary differential equations on the half-line, Ann. scuola norm. sup. Pisa cl. sci., 2, 265-286, (1975) · Zbl 0326.34034  Toland, J.F, Global bifurcation for Neumann problems without eigenvalues, J. differential equations, 44, 82-110, (1982) · Zbl 0455.34015  Turner, R.E.L, Superlinear Sturm-Liouville problems, J. differential equations, 13, 157-171, (1973) · Zbl 0272.34031
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