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} {\mathbb 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 \({\mathbb R}^n\) with radius \(R>0.\)


35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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