Let $\Omega$ a smooth and bounded domain, $f\in L^2(\Omega)$ and $L$ an uniform elliptic operator in divergence form. Consider the problem: $$(1) \left\{ \begin{array}{ll} Lu=f \hspace{.4cm} \mbox{ in } \Omega,\\ u=0 \hspace{.4cm} \mbox{ in } \partial\Omega. \end{array} \right.$$ QUESTION: There exist uniform elliptic operator in divergence form $f$ such that the problem $(1)$ has no solution for every $f\neq 0$???
My attempt: The idea is using Fredhölm Alternative, considering the adjoint homogeneous problem: $$(H) \left\{ \begin{array}{ll} Lu=0 \hspace{.4cm} \mbox{ in } \Omega,\\ u=0 \hspace{.4cm} \mbox{ in } \partial\Omega. \end{array} \right.$$ Fredhölm states that if $u\in A:=\left \{ u\in H_0^1(\Omega) | u \mbox{ is solution of } (H)\right \}$ then u is solution of $(1)$ if and only if: $$(u,f)_{L^2(\Omega)}=\int_\Omega uf=0, i.e. f\in A^\perp.$$ So I must show an operator $L$ such that for every $u\in A$ and $f\in L^2(\Omega)\backslash \{0\}$ occurs: $$(u,f)_{L^2(\Omega)}\neq 0.$$
I understand the theory but I don't know which operator I must use to prove the last statement. Every comment and idea would be so appreciated, thanks ♥
Let $\lambda$ be an eigenvalue of $-\Delta$, i.e., there is $w\ne0$, $w\in H^1_0(\Omega)$ such that $$ -\Delta w = \lambda w $$ in the weak sense. Then set $L:= -\Delta - \lambda$. Then $w\in A$ and $A^\perp$ is not the whole space.