= Solution
The <Lax-Milgram theorem> states that a <bounded bilinear form> $a$ on a real <Hilbert space> $V$, satisfying $a(v,v)\ge\alpha\|v\|_V^2$ for some $\alpha>0$, has for each $F\in V^*$ a unique $u\in V$ with $a(u,v)=F(v)$ for all $v$. Moreover $\|u\|_V\le\|F\|/\alpha$.
For a <compact operator> $K$ on a <Hilbert space>, the <Fredholm alternative> says that $I-\lambda K$ has closed range, <finite-dimensional vector space>[finite-dimensional] <null space>[kernel] and cokernel of equal dimension, and
$$
\operatorname{Ran}(I-\lambda K)=\ker(I-\lambda K^*)^\perp\qquad(\lambda\in\mathbb R).
$$
Thus a zero <null space>[kernel] gives unique solvability for every right-hand side; a nonzero <null space>[kernel] gives solvability exactly under the adjoint <orthogonality> conditions, and then all solutions differ by a <null space>[kernel] <vector>.
Take $V=H_0^1(\Omega)$. <Gårding's inequality>[Gårding's inequality] supplies constants $a>0$, $b\ge0$ with $B[v,v]\ge a\|v\|_{H^1}^2-b\|v\|_2^2$. Therefore for $\lambda\le-b$ the shifted form $B_\lambda[u,v]=B[u,v]-\lambda(u,v)_{L^2}$ is coercive. It is <bounded> for every fixed $\lambda$, and $v\mapsto(f,v)$ is <bounded> on $V$. Lax-Milgram gives \b[unique weak solvability for $\lambda\le-\gamma$, with $\gamma=b$]. The inequality itself follows, for example, by absorbing the drift cross-term $\|b\|_\infty\|\nabla v\|_2\|v\|_2$ using Young's inequality and allowing a lower $L^2$ correction for $c$.
For the <compact inverse and adjoint of a coercive elliptic operator>, under the additional zero-shift coercivity assumption, define $Gg\in V$ by $B[Gg,v]=(g,v)$ for all $v\in V$, and let $K$ be this solution viewed in $L^2$. The solution operator is linear and <bounded> into $V$, and the <Rellich-Kondrachov theorem>[Rellich-Kondrachov] <compact embedding> $V\hookrightarrow L^2$ makes $K$ compact. Rewriting the weak equation as $B[u,v]=(f+\lambda u,v)$ gives
$$
\boxed{(I-\lambda K)u=Kf.}
$$
Conversely an $L^2$ solution of this operator equation lies in $V$, because it equals $K(f+\lambda u)$, and satisfies the weak <partial differential equation>[differential equation].
For constant drift coefficients and zero <Dirichlet boundary condition>[Dirichlet] <Sobolev trace>[traces], <integration by parts> gives $\int b_iD_i v\,u=-\int b_i vD_i u$. Symmetry of the principal coefficients therefore makes $B^*[u,v]=B[v,u]$ the unshifted weak form of the displayed formal adjoint. \b[The shifted adjoint equation requires]
$$
\boxed{B^*[u,v]-\lambda(u,v)=(f,v)\qquad(v\in V).}
$$
The <spectral shift in an adjoint weak formulation> is essential: the printed sentence omitting it is false for nonzero $\lambda$. For instance take a smooth ball, $L=-\Delta+1$, any nonzero $u\in C_c^\infty(\Omega)$ and $f=-\Delta u+u$. The printed unshifted identity holds, but $(L^*-\lambda)u=f-\lambda u\ne f$ whenever $\lambda\ne0$. This counterexample also meets zero-shift coercivity and constant-drift assumptions. The corrected identity above is used for the remainder of the argument.
Let $G_*g\in V$ solve $B^*[G_*g,v]=(g,v)$, which exists by Lax-Milgram because the diagonal coercivity is unchanged. For $p,q\in L^2$,
$$
(Kp,q)=B[Gp,G_*q]=(p,G_*q).
$$
Thus $G_*$, viewed as an $L^2$ operator, is exactly $K^*$. Applying this inverse to the corrected shifted adjoint equation gives
$$
\boxed{(I-\lambda K^*)u=K^*f,}
$$
and the converse follows from its range lying in $V$, just as before.
Finally let the original homogeneous equation have a nonzero solution. Its parameter is nonzero, since the unshifted problem is coercive. Fredholm gives solvability precisely when $(Kf,w)=0$ for every $w\in\ker(I-\lambda K^*)$. Such a $w$ satisfies $K^*w=w/\lambda$, so $(Kf,w)=(f,w)/\lambda$. Hence the necessary and sufficient condition is
$$
\boxed{\int_\Omega fw\,dx=0\quad\text{for every }w\in H_0^1(\Omega)\text{ satisfying }(L^*-\lambda)w=0\text{ weakly}.}
$$
When this holds, the solutions are an <affine subspace>[affine translate] of the original <finite-dimensional vector space>[finite-dimensional] <null space>[homogeneous solution space]. The compatibility space is the adjoint <null space>[kernel]; a drift term does not generally permit replacing it by the original <null space>[kernel].
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