On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations

Tadie

Abstract: For $f\in C([0,\infty)) \cap C^1((0, \infty))$ and $b > 0$, existence and uniqueness of radial solutions $u = u(r)$ of the problem $\triangle u + f(u_+) = 0$ in ${\Bbb R}^n \ (n > 2), \, u(0) = b$ and $u'(0) = 0$ are well known. The uniqueness for the above problem with boundary conditions $u(R) = 0$ and $u'(0) = 0$ is less known beside the cases where $\lim_{r\to \infty}u(r) = 0$. It is our goal to give some sufficient conditions for the uniqueness of the solutions of the problem $D_\alpha u + f(u_+) = 0 \ (r > 0), u(\rho) = 0$ and $u'(0) = 0$ based only on the evolution of the functions $f(t)$ and ${d \over dt}{f(t) \over t}$.

Keywords: semilinear elliptic equations, comparison results for nonlinear differential equations

Classification (MSC2000): 35J65, 34B15

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