Nonuniqueness of strong solution

If we assume u\in W^{2,p}_{loc}(\Omega)\cap C^0(\bar{\Omega}), where p<n, then the uniqueness of Dirichlet problem will fails. See the following example

\begin{cases}\displaystyle \Delta u+\left(-1+\frac{n-1}{1-\lambda}\right)D_{ij}u=0\\ u(x)=1\quad \text{ on }\partial B_1(0)\end{cases}

where n>2(2-\lambda)>2. One can verify that u(x)=1 and u(x)=|x|^\lambda\in W^{2,2}(B_1) are two solutions.

 

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