Uniqueness of positive radial solutions of Δu+f(u)=0 in ℝn

We prove a uniqueness result for the positive solution of A« + f(u) = 0 in R" which goes to 0 at oo . The result applies to a wide class of nonlinear functions /, including the important model case f(u) = -u + up , 1 < p < (n + 2)l(n 2). The result is proved by reducing to an initial-boundary problem for the ODE u" + (n l)/r + f(u) = 0 and using a shooting method.

[1]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[2]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[3]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[4]  R. Nussbaum,et al.  Uniqueness and nonuniqueness for positive radial solutions of Δu+f(u,r)=0 , 1985 .

[5]  W. Ni.  On the positive radial solutions of some semilinear elliptic equations on ℝn , 1982 .

[6]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[7]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[8]  L. A. Peletier,et al.  Ground states of −Δu=f(u) and the Emden-Fowler equation , 1986 .

[9]  Lambertus A. Peletier,et al.  Uniqueness of non-negative solutions of semilinear equations in Rn , 1986 .

[10]  Z. Nehari,et al.  ON A NONLINEAR DIFFERENTIAL EQUATION ARISING IN NUCLEAR PHYSICS , 1963 .

[11]  Lambertus A. Peletier,et al.  Uniqueness of positive solutions of semilinear equations in ℝn , 1983 .

[12]  C. V. Coffman.  Uniqueness of the ground state solution for Δu−u+u3=0 and a variational characterization of other solutions , 1972 .

[13]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[14]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, II existence of infinitely many solutions , 1983 .

[15]  L. E. Scriven,et al.  Molecular theory of fluid interfaces , 1976 .

[16]  Yasushi Takahashi.  The structure of the nucleon core by the Hartree approximation , 1961 .

[17]  X. Garaizar.  Existence of Positive Radial Solutions for Semilinear Elliptic Equations in the Annulus , 1987 .