A Global multiplicity result for a very singular critical nonlocal equation
Keywords
Fractional Laplacian, very singular nonlinearity, variational method, Hölder regularityAbstract
In this article we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (-\Delta)^s u = u^{-q} + \lambda u^{{2^*_s}-1}, \quad u> 0 \quad \text{in } \Omega,\quad u = 0 \quad \mbox{in } \mathbb R^n \setminus\Omega, \tag{ P$_\lambda$} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s$, $ s \in (0,1)$, $ \lambda > 0$, $q> 0$ satisfies $q(2s-1)< (2s+1)$ and $2^*_s=2n/(n-2s)$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(\rom{P}_\lambda)$ in $X_0$ when $\lambda \in (0,\Lambda)$ and no solution when $\lambda> \Lambda$, where $\Lambda> 0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to (P$_\lambda)$ is in $C^\alpha(\mathbb R^n)$ with $\alpha=\alpha(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.References
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