  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 $t>0$.
