Conjecture on the ground state and principal eigenvalue of the Gierer–Meinhardt core problem

Let $S>0$ and consider the core problem eqref{eq:core}, whose ground state solution is denoted by $(U(\rho;S),V(\rho;S))$, with $mu(S)$ the associated eigenvalue function. A solution has the required positivity property when $V,U>0$ for $\rho>0$. Ground-state and eigenvalue conjecture. There exists a unique value of $S_\star>0$ such that eqref{eq:core} admits a ground state solution with $V,U>0$ for $\rho>0$ and $mu(S_\star)=0$. Moreover, $mu(S)>0$ and $mu^{\prime\prime}(S)<0$ for every $0<S<S_\star$. The existence of these core solutions and the stated properties of $mu(S)$ are not rigorously established and are left as an open problem; they underpin the subsequent asymptotic and stability analysis of the localized three-dimensional spot patterns.