The effect of domain shape on the number of positive solutions of certain nonlinear equations.(English)Zbl 0662.34025

The author considers how the shape of the bounded domain $$\Omega$$ $$(\Omega \in R^ m$$, $$m>1)$$ affects the number of positive solution to the problem (i) $$-\Delta u=\lambda f(u)$$ in $$\Omega$$, $$u=0$$ on $$\partial \Omega$$, where $$f\in C^ 1[{\mathbb{R}},{\mathbb{R}}]$$. For instance, he proves that if $$f(u)=\exp u$$, then there are contractible domains $$\Omega$$ for which equation (i) has large number of solutions. The case $$f(u)=u^ p$$ is also under particular attention. The conditions assuring that the problem (i) (with $$f(u)=u^ p)$$ has a unique positive, nondegenerate solution are given. A lot of examples illustrate the obtained results.
Reviewer: D.Bobrowski

MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
Full Text:

References:

 [1] Amann, H, Multiple fixed points of asymptotically linear maps, J. funct. anal., 17, 174-213, (1974) · Zbl 0287.47037 [2] Amann, H; Laetsch, T, Positive solutions of convex nonlinear eigenvalue problems, Indiana univ. math. J., 23, 1069-1076, (1974) [3] Amick, C; Toland, J, Nonlinear elliptic eigenvalue problems on an infinite stripglobal theory of bifurcation and asymptotic bifurcation, Math. ann., 262, 313-342, (1983) · Zbl 0489.35067 [4] Aris, R, () [5] Bahri, A; Coron, J.M, Sur une équation elliptique non linéaire avec l’exposant critique de Sobolev, C. R. acad. sci. Paris, 301, 345-348, (1985) · Zbl 0601.35040 [6] Berger, M, Nonlinearity and functional analysis, (1977), Academic Press New York [7] Crandall, M; Rabinowitz, P.H, Bifurcation from simple eigenvalues, J. funct. anal., 8, 321-340, (1971) · Zbl 0219.46015 [8] Crandall, M; Rabinowitz, P.H, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. rational mech. anal., 58, 207-218, (1975) · Zbl 0309.35057 [9] Dancer, E.N, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. differential equations, 37, 404-437, (1980) · Zbl 0417.34042 [10] Dancer, E.N, On positive solutions of some pairs of differential equations II, J. differential equations, 60, 236-258, (1985) · Zbl 0549.35024 [11] Dancer, E.N, Remarks on multiple solutions of nonlinear equations, (), 25-31 · Zbl 0466.35038 [12] Dancer, E.N, On non-radially symmetric bifurcation, J. London math. soc., 20, 287-292, (1979) · Zbl 0418.35015 [13] Dancer, E.N, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, (), 429-452 · Zbl 0572.35040 [14] Dancer, E.N, On the indices of fixed points of mappings in cones and applications, J. math. anal. appl., 91, 131-151, (1983) · Zbl 0512.47045 [15] Dancer, E.N, Multiple fixed points of positive mappings, J. reine angew. math., 37, 46-66, (1986) · Zbl 0597.47034 [16] Dancer, E.N, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. ann., 272, 421-440, (1985) · Zbl 0556.35001 [17] Dancer, E.N; Schmitt, K, On positive solutions of semilinear elliptic equations, (), 445-452 · Zbl 0661.35031 [18] Gelfand, E.M, Some problems in the theory of quasilinear equations, Amer. math. soc. transl., 29, 295-381, (1963), (2) · Zbl 0127.04901 [19] Gidas, B; Ni, W; Nirenberg, L, Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020 [20] Gidas, B; Spruck, J, Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003 [21] Gidas, B; Spruck, J, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 883-901, (1981) · Zbl 0462.35041 [22] Gilbarg, D; Trudinger, N, Elliptic partial differential equations of second order, (1977), Springer Berlin · Zbl 0361.35003 [23] Grisvard, P, Elliptic problems in nonsmooth domains, (1985), Pitman London · Zbl 0695.35060 [24] Hale, J; Vegas, J, A nonlinear parabolic equation with varying domain, Arch. rational mech. anal., 86, 99-123, (1984) · Zbl 0569.35048 [25] Hartman, P; Wintner, A, On the local behaviour of solutions of non-parabolic partial differential equations. III. approximation by spherical harmonics, Amer. J. math., 77, 453-474, (1955) · Zbl 0066.08001 [26] Hess, P, On multiple solutions of nonlinear elliptic eigenvalue problems, Comm. partial differential equations, 6, 951-961, (1981) · Zbl 0468.35073 [27] Joseph, D.J; Lungren, T, Quasilinear Dirichlet problems driven by positive sources, Arch. rational mech. anal., 49, 241-249, (1973) · Zbl 0266.34021 [28] Keylitz, B; Kuiper, H, Bifurcation resulting from changes in domain in a reaction diffusion equation, J. differential equations, 47, 378-405, (1983) · Zbl 0519.35040 [29] Kinderlehrer, D; Stampacchia, G, An introduction to variational inequalities, (1980), Academic Press New York · Zbl 0457.35001 [30] Lloyd, N, Degree theory, (1978), Cambridge Univ. Press Cambridge · Zbl 0367.47001 [31] Matano, H; Mimara, M, Pattern formulation in competitive-diffusion systems in non-convex domains, Publ. res. inst. math. sci., 19, 1049-1079, (1983) · Zbl 0548.35063 [32] Rauch, J; Taylor, M, Potential and scattering theory in wildly perturbed domains, J. funct. anal., 18, 27-59, (1975) · Zbl 0293.35056 [33] Saut, J; Teman, R, Generic properties of nonlinear boundary-value problems, Comm. partial differential equations, 4, 293-319, (1979) · Zbl 0462.35016 [34] Schaeffer, D, Nonuniqueness in the equilibrium shape of a confined plasma, Comm. partial differential equations, 2, 587-600, (1977) · Zbl 0371.35017 [35] Stummel, F, Perturbation of domains in elliptic boundary value problems, (), 110-135 [36] Sweers, G, Maximum estimates for a semilinear elliptic eigenvalue problem, (1986), preprint, Delft [37] Vainberg, M.M, Variational methods for the study of nonlinear operators, (1964), Holden-Day San Fancisco · Zbl 0122.35501 [38] Vegas, J, Bifurcation caused by perturbing the domain in an elliptic equation, J. differential equations, 48, 189-222, (1983) · Zbl 0465.35075 [39] Babuska, I; Vyborny, R, Continuous dependence of the eigenvalues on the domain, Czech. math. J., 15, 169-178, (1965) · Zbl 0137.32302 [40] Courant, R; Hilbert, D, () [41] Garabedian, P; Schiffer, M, Convexity of domain functionals, J. analyse math., 2, 281-368, (1952) · Zbl 0052.33203 [42] Grigorieff, R, Kompakte einbettungen in sobolewschen raümen, Math. ann., 197, 71-85, (1972) · Zbl 0222.35002 [43] Mignot, J; Murat, F; Puel, J, Variation d’un point the retournement par rapport au domaine, Comm. partial differential equations, 4, 1236-1297, (1979) · Zbl 0422.35039 [44] Necas, J, LES méthodes directes en théorie de l’équations elliptiques, (1967), Academia Prague [45] Stummel, F, Perturbation theory for Sobolev spaces, (), 5-49 · Zbl 0358.46027
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.