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Uniqueness of ground states for quasilinear elliptic equations. (English) Zbl 0979.35049
The authors give a condition for the uniqueness of ground states (nonnegative nontrivial $C^1$ distribution solution which tends to zero at $\infty$) of the quasilinear elliptic equation $$\text{div}(|Du|^{m-2}Du) =f(u)\quad \text{ in} {\Bbb R}^n,\quad n>m>1. \tag $*$ $$ Precisely, $(*)$ admits at most one radial ground state if, for some $b>0,$ $f\in C(0,\infty),$ with $f(u)\leq 0$ on $(0,b]$ and $f(u)>0$ for $u>b;$ $f\in C^1(b,\infty),$ with $g(u)=uf'(u)/f(u)$ non-increasing on $(b,\infty).$ In addition, it is considered also uniqueness of radial solutions of the homogeneous Dirichlet-Neumann free boundary problem for the equation $(*)$ with $u>0$ in $B_R,$ $u=\partial u/\partial n=0 $ on $\partial B_R, $ where $B_R$ is an open ball in ${\Bbb R}^n$ with radius $R>0.$

35J60Nonlinear elliptic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
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