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Keywords:
spherically symmetric solution; trajectory of the solution; со-limit point of the trajectory; asymptotic formula; antitone and contractive operator; zero of the solution; Klein-Gordon equation; global behavior
spherically symmetric solution; trajectory of the solution; со-limit point of the trajectory; asymptotic formula; antitone and contractive operator; zero of the solution; Klein-Gordon equation; global behavior
Summary:
In the paper it is shown that each solution $u(r,\alpha)$ ot the initial value problem (2), (3) has a finite limit for $r\rightarrow \infty$, and an asymptotic formula for the nontrivial solution $u(r,\alpha)$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u(r,\bar{\alpha})$, $u(r,\hat{\alpha})$.
References:
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[2] L. Erbe K. Schmitt: On radial solutions of some semilinear elliptic equations. Differential and Integral Equations, Vol. 1 (1988), 71 - 78. MR 0920490
[3] J. Chauvette F. Stenger: The approximate solution of the nonlinear equation $\Delta u = u - u^3$. J. Math. Anal. Appl. 51 (1975), 229-242. DOI 10.1016/0022-247X(75)90155-9 | MR 0373320
[4] G. H. Ryder: Boundary value problems for a class of nonlinear differential equations. Pacific J. Math. 22 (1967), 477-503. DOI 10.2140/pjm.1967.22.477 | MR 0219794 | Zbl 0152.28303
[5] G. Sansone: Su un'equazione differenziale non lineare della fisica nucleare. Istituto Nazionale di Alta Matem. Sympozia Mathematica, Vol. VI, (1970).
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