Liouville theorem for semilinear elliptic inequalities involving the fractional Hardy operators

Abstract:

In this paper, we give a full classification of the nonexistence of positive weak solutions to the following semilinear elliptic inequality \begin{equation*} (-\Delta )^s u+\frac {\mu }{|x|^{2s}} u\geq Q u^p \end{equation*} in a bounded punctured domain or in an exterior domain, where $p>0$, $\mu \geq \mu _0$ and $\liminf Q(x)|x|^{-\theta }>0$ for some $\theta \in \mathbb {R}$, $-\mu _0$ is the best constant in the fractional Hardy inequality. Our work covers the important case $\theta \leq -2s$ for which all the existence methods do not apply to get the Liouville type theorems.

In the punctured domain, Brézis-Dupaigne-Tesei proved the nonexistence of positive solutions of the equation given above when $s=1$, $\mu \in [\mu _0,0)$ and $Q=1$. When $s\in (0,1)$, we extend this nonexistence result to a general setting via a new method. We also provide a critical exponent $p^\#_{\mu , \theta }$ depending on the parameters $\mu \in [\mu _0,0)$, and $\theta \in \mathbb {R}$. In the supercritical case, $p>p^\#_{\mu , \theta }$, we obtain the nonexistence of positive solutions for the equation given above, while in the critical case $p=p^\#_{\mu , \theta }$ this nonexistence holds true if $p^\#_{\mu , \theta }>1$. When $p=p^\#_{\mu , \theta }\in (0,1)$, positive solutions can be constructed provided that we have an appropriate upper bound of $Q$ near the origin additionally.

In an exterior domain, we provide the Serrin’s type critical exponent $p^*_{\mu , \theta }$ depending on the parameters $\mu \geq \mu _0$, $\theta \in \mathbb {R}$, and then we obtain the nonexistence of positive solutions in the subcritical case: $p\in (0,p^*_{\mu , \theta })$. The nonexistence in the critical case $p=p^*_{\mu , \theta }>1$ and the existence of positive solutions in the critical case $p=p^*_{\mu , \theta }\in (0,1)$ are established under additional assumptions on the upper bound of $Q$ at infinity.

Our methods are self-contained and new. The main ideas are established in the section after the first theorem in the article for punctured domains, and after the third theorem in the article for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste: An initial asymptotic behavior rate at the origin or at infinity could be derived by the two fundamental solutions, we then improve this rate by using the interaction with the nonlinearity, based on the imbalance between the Hardy operator and nonlinearity. By repeating this process a finite number of times, a contradiction will be deduced from the nonexistence for the related non-homogeneous fractional Hardy problem. This process allows us to obtain the nonexistence for the fractional Hardy problem with larger ranges of $\theta$ and $p$. Our study covers all possible ranges, and our results are optimal.