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In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H\"older continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} \Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $\alpha$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $\beta$-H\"older continuous for $1<\alpha<2$ and $0<\beta<1$. For $G\equiv0$ and $H(x)=x^\beta$, in Mytnik (2002) a weak solution to the above SPDE was constructedand the pathwise uniqueness of the solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of $\alpha$ and $\beta$. In particular, for $\alpha\beta=1$, where the solution to the above equation is the density of a super-Brownian motion with $\alpha$-stable branching (see also Mytnik (2002)), our result leads to its pathwise uniqueness for $1<\alpha<\sqrt{5}-1$. The local H\"older continuity of the solution is also obtained in this paper for fixed time $t>0$.

Pathwise uniqueness for an SPDE with Hölder continuous coefficient driven by $\alpha $-stable noise