Abstract
The purpose of this paper is to describe some recent joint work of V. Coti Zelati and the author on semilinear elliptic partial differential equations on R n [1]. This research is in part an outgrowth of earlier work on homoclinic solutions of Hamiltonian systems of ordinary differential equations [2].
This research was sponsored in part, by the National Science Foundation under Grant #MCS-8110556 and the U.S. Army Research Office under Contract #DAAL03-87-K-0043.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Coti Zelati, V. AND P. H. Rabinowitz, Homociinic type solutions for a class of semilinear elliptic equations on R n, Comm. Pure Appl. Math, (to appear).
Coti Zelati, V. AND P. H. Rabinowitz, Homociinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. (to appear).
McLEOD, K. AND J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Nat. Acad. Sci. U.S.A., 28 (1981), pp. 6592–6595.
Synge, J. L., Oil a certain nonlinear differential equation, Proc. Roy. Irish Acad., 62 (1961), pp. 17–41.
Nehari, Z., On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish Acad., 62 (1963), pp. 117–135.
Berger, M. S., On the existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. Funct. Anal., 9 (1972), pp. 249–261.
Coffman, C. F., Uniqueness of the ground state solution for ▵u—u+u3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal., 46 (1972), pp. 81–85.
Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1979), pp. 149–162.
Berestycki, H. AND P. L. LIONS, Nonlinear scalar held equations, I. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), pp. 313–345.
Lions, P. L., The concentration compactness principle in the calculus of variations. The locally compact case. Parts I &II., Analyse Nonlin., 1 (1984), pp. 109–145, 223-283.
LionS, P. L., The concentration-compactness principle in the calculus of variations, Rev. Mat. Iberoamericana, 1 (1985), pp. 145–201.
DinG, W.-Y. AND W.-M. NI, On the existence of a positive entire solution of a semilinear elliptic equation, Arch. Rat. Mech. Anal., 91 (1986), pp. 283–308.
Floer, A. AND A. Weinstein, Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential, J. Funct. Anal., 69 (1986), pp. 397–408.
Angenent, S., The shadowing lemma for elliptic PDE, Dynamics of Infinite Dimensional Systems (1987.), (S. N. Chow and J. K. Hale, eds) Springer-Verlag.
NI, W. M., Some aspects of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States (1988), (W. M. Ni, L. A. Peletier, and J. Serrin, eds.), Springer Verlag.
OH, Y.-G., Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. Partial Diff. Eq., 13 (1988), pp. 1499–1519.
OH, Y.-G., Corrections to “Existence of semi-classical bound state of nonlinear Schrödinger equations with potentials of the class (V)a ”, Comm. Partial Diff. Eq., 14 (1989), pp. 833–834.
LI, Y., Remarks on a semilinear elliptic equation on R n, J. Diff. Eq., 74 (1988), pp. 34–49.
OH, Y.-G., Stability of semiclassical bound states of nonlinear Schrödinger equations with potential, Comm. Math. Phys., 121 (1989), pp. 11–33.
NI, W.-M., Recent progress in semilinear elliptic equations, RIMS Kokyuroku, (T. Suzuki, ed., Kyoto Univ), 679 (1989), pp. 1–39.
OH, Y.-G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), pp. 223–253.
Bahri, A. AND Y. Y. LI, On a minimax procedure for the existence of a positive solution for certain scalar held equations in R N, Rev. Mat. Iberoamericana (to appear).
Yu, L. S., Positive decaying solutions of nonlinear elliptic problems, thesis, University of Alberta, Fall 1990.
Alema, S. AND Y. LI, Existence of solutions for semilinear elliptic equations with indennite linear part, preprint.
Rabinowitz, P. H., A note on a semilinear elliptic equation on R n. Nonlinear Analysis, a tribute in honor of Giovanni Prodi, Quaderni Scuola Normale Superiori, Pisa.
Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z.A.M.P. (to appear).
Ambrosetti, A. AND P. H. Rabinowitz, Duai variationai methods in critical point theory and applications, J. Funct. Anal., 14 (1973), pp. 349–381.
Smale, S., Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (edited by S. S. Cairns), Princeton Univ. Press (1965), pp. 63–80.
Moser, J., Stable and Random Motions in Dynamical Systems, Princeton Univ. Press (1973).
Kirchgraber, U. And D. Stoffer, Chaotic behavior in simple dynamical systems, Siam Review, 32 (1990), pp. 424–452.
Chang, K. C. And J.-Q. Liu, A remark on the homoclinic orbits for Hamiltonian systems, preprint.
RAbinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Reg. Conf. Series in Math., Amer. Math. Soc, Providence, RI, 65 (1986).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rabinowitz, P.H. (1993). Multibump Solutions of a Semilinear Elliptic PDE on Rn . In: Ni, WM., Peletier, L.A., Vazquez, J.L. (eds) Degenerate Diffusions. The IMA Volumes in Mathematics and its Applications, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0885-3_12
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0885-3_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6935-9
Online ISBN: 978-1-4612-0885-3
eBook Packages: Springer Book Archive