Counter example to unique solvability of Dirichlet Problem

Question

In Gilbarg and Trudinger, Theorem 6.14 states that for a uniformly elliptic operator $L=A^{ij}D_{ij}+B^iD_i + c$ the Dirichlet problem

$ Lu=f $

has a unique solution $u$ for every $f$ under some suitable assumptions.

One of these assumptions is that $c \leq 0$, and the book says that there are counter examples where the above does not hold for $c \geq 0$. So, what would be such a counter example, and why is it a counter example?

Thanks in advance.

Answer 1

The classic examples are eigenvalues of the Laplacian: if $\Delta v = \lambda v$ for some nontrivial $v$ with zero Dirichlet condition, then any constant multiple $u=Cv$ is a solution of the Dirichlet problem $$\Delta u - \lambda u = 0.$$

Thus for an explicit counterexample you can take $\Omega = (0,\pi),$ $A^{ij}=\delta^{ij},$ $B^i = 0,$ $c = 1,$ $f = 0,$ which admits the family of solutions $$u(x) = C \sin(x),\quad C\in\mathbb R.$$