| Abstract: |
| This talk is concerned with the uniqueness of radially symmetric ground state solutions to the double phase elliptic problem
$$
\mbox{div}\big( a(|x|)|\nabla v|^{q-2}\nabla v+b(|x|)|\nabla v|^{p-2}\nabla v \big)+\mathsf f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,\eqno{(P)}
$$
where $a$ and $b$ are weights satisfying some growth assumptions. For simplicity in this talk we will treat the case of $a(|x|)=|x|^{\tilde\theta}$, and $b(|x|)\equiv 1$, namely
\begin{equation}\label{eq2}
\begin{gathered}
\big(r^{\theta-1}\phi_q( u`)+r^{n-1}\phi_p(u`))`+r^{n-1}f(u)=0\,,\quad r>0\,,\
u`(0)=0\,,\quad\lim\limits_{r\to+\infty}u(r)=0\,,
\end{gathered}
\end{equation}
where $\theta=n+\tilde\theta>1$, $\phi_s(t)=|t|^{s-2}t$, $s>1$, $n>q>p>1$ and $`=\frac{d}{dr}$ .
\medskip
Regarding the nonlinearity $f\in C(\mathbb R)$, we assume the following
\begin{enumerate}
\item[$(f_1)$]$f$ is odd, $f(0)=0$, and there exist $0\max\left\{\frac{\theta}{nq}-\frac{1}{n},\frac{1}{p}-\frac{1}{n}\right\}$, \quad for $s>\beta,$
\end{enumerate}
Regarding the weight $r^{\theta-1}$ we assume
\begin{enumerate}
\item[$(w_1)$] $\displaystyle\theta\geq n,\quad \frac{\theta-n}{q-p} |
|