## Uniqueness of the ground state solution for $$\Delta u - u + u^3=0$$ and a variational characterization of other solutions.(English)Zbl 0249.35029

### MSC:

 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations
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### References:

 [1] Berger, M. S., Stationary states for a nonlinear wave equation. J. Math. Physics 11, 2906–2912 (1970). · Zbl 0201.12602 [2] Betts, D. D., H. Schiff, & W. B. Strickfaden, Approximate gnsolution of a nonlinear field equation. J. Math. Physics 4, 334–338 (1963). · Zbl 0128.46003 [3] Coffman, C. V., On the equat [4] Coffman, C. V., Spectral theory of monotone Hammerstein operators. Pac. J. Math. 36, 303–322 (1971). · Zbl 0212.46803 [5] Coffman, C. V., A minimum-maximum principle for a class of non-linear integral equations. J. d’Analyse Math. 22, 391–419 (1969). · Zbl 0179.15601 [6] Coffman, C. V., On the positive solutions of boundary-value problems for a class of nonlinear differential equations. J. Diff. Eq. 3, 92–111 (1967). · Zbl 0152.08603 [7] Coffman, C. V., & A. J. Das, A class of eigenvalues of the fine structure constant and internal energy obtained from a class of exact solutions of the combined Klein-Gordon-Maxwell-Einstein field equations. J. Math. Physics 8, 1720–1735 (1967). · Zbl 0157.32204 [8] Conner, P. E., & E. E. Floyd, Fixed point free involutions and equivariant maps. Bull. Amer. Math. Soc. 66, 416–441 (1960). · Zbl 0106.16301 [9] Dugundji, J., Topology. Boston: Allyn and Bacon 1966. [10] Finklestein, R., R. Lelevier, & M. Ruderman, Nonlinear spinor fields. Phys. Review 83, 326–332 (1951). · Zbl 0043.21603 [11] Friedman, A., Partial Differential Equations. New York: Holt. Rinehart and Winston (1969). · Zbl 0224.35002 [12] Hartman, P., Ordinary Differential Equations. New York: Wiley (1964). · Zbl 0125.32102 [13] Kolodner, I. I., Heavy rotating string–a nonlinear eigenvalue problem. Comm. Pure and Appl. Math. 8, 395–408 (1955). · Zbl 0065.17202 [14] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach (1963). · Zbl 0121.42701 [15] Moser, J., A sharp form of an inequality by N. Trudinger. Indiana University Mathematics Journal 20, 1077–1092 (1971). · Zbl 0203.43701 [16] Mostow, G. D., Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Inst. des Hautes Études Scient., Publ. Math., No 34 (1968). · Zbl 0189.09402 [17] Nehari, Z., On a nonlinear differential equation arising in nuclear physics. Proc. Royal Irish Academy, 62, 117–135 (1963). · Zbl 0124.30204 [18] Nirenberg, L., Remarks on strongly elliptic partial differential equations, Comm. Pure and Appl. Math. 8, 648–674 (1955). · Zbl 0067.07602 [19] Pólya, G., & G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Princeton: Princeton University Press (1951). · Zbl 0044.38301 [20] Robinson, P. D., Extremum pri · Zbl 0205.09901 [21] Ryder, G. H., Boundary value problems for a class of nonlinear differential equations. Pac. J. Math. 22, 477–503 (1967). · Zbl 0152.28303 [22] Synge, J. L., On a certain nonlinear differential equation. Proc. Royal Irish Academy 62, 17–42 (1961). · Zbl 0104.31501 [23] Teshima, R. K., M. Sc. Thesis. University of Alberta, Edmonton, Alberta, Canada, (1960). [24] Weiss, S., Nonlinear eigenvalue problems. Ph. D. Dissertation, University of Chicago, August 1969. [25] Wolkowisky, J. H., Existence of buckled states of circular plates. Comm. Pure and Appl. Math. 20, 549–560 (1967). · Zbl 0168.45206 [26] Sansone, G., Su un’equazione differenziale non lineare della fisica nucleare, Symposia Mathematica, vol. VI. Istituto Nazionale di Alta Matematica, Bologna (1970).
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