## 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 $$R>0.$$

### MSC:

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