| Abstract: |
| (based on a joint work with Profs. Carmen Cort\'azar and Pilar Herreros)
This talk is concerned with the uniqueness of radially symmetric ground state solutions to the double phase elliptic problem
$$
\mbox{div}\big( a(|x|)|\nabla v|^{q-2}\nabla v+b(|x|)|\nabla v|^{p-2}\nabla v \big)+\mathsf f(v)=0\,,\lim_{|x|\to+\infty}v(x)=0,\, x\in\mathbb R^n, (P)
$$
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\,, r>0\,,
u'(0)=0\,,\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 $'=\frac{d}{dr}$ .
Regarding the nonlinearity $f\in C(\mathbb R)$, we assume the following
\item[$(f_1)$]$f$ is odd, $f(0)=0$, and there exist $0\lt b\lt \beta\lt \infty$ such that $f(s)>0$ for $s>b$,
$f(s)\le 0$, $f(s)\not\equiv 0$ for $s\in[0,b]$, $F(\beta)=0$, where
$F(s):=\int_0^sf(t)dt$.
\item[$(f_2)$] $f$ is continuously differentiable in $(0,\infty)$, $f'\in L^1(0,1)$.
\item[$(f_3)$] $f(x)/\phi_q(x)$ is an increasing function of $x$ for $x>b$.
\item[$(f_4)$] $\Bigl(\displaystyle\frac{F}{f}\Bigr)'(s)>\max\left\{\frac{\theta}{nq}-\frac{1}{n},\frac{1}{p}-\frac{1}{n}\right\}$, \quad for $s>\beta,$
Regarding the weight $r^{\theta-1}$ we assume
\item[$(w_1)$] $\displaystyle\theta\geq n,\quad \frac{\theta-n}{q-p}\lt 1$. |
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