I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.

There is a linear operator $L:{D}(L) \to V^*$ where $D(L) \subset V$ is a linear subspace of $V$ and $D(L)$ is a Hilbert space with an inner product $(\cdot,\cdot)_{D(L)}$ (I don't think it is a closed subspace of $V$). $L$ is monotone in the sense that $$\langle Lu, u \rangle_{V^*, V} \geq 0.$$ We have linear operators $A:V \to V^*$ and $B: V \to H$ both of which are bounded and furthermore we have $$\langle Au,u \rangle_{V^*,V} \geq C_0\lVert{u}\rVert_{V}^2$$ so $A$ is monotone. $B$ is not monotone though. The problem is: find $u \in D(L)$ such that for $f \in V^*$ $$Lu + Au + Bu = f$$ holds in $V^*$.

So my problem is that $B$ is not monotone and I do not want to restrict the size of the bound on $B$. So I don't want to make an assumption like "Let us assume $\lVert B \rVert$ is small enough, then we have well-posedness...".

Does anyone know how to solve such problems and if they can refer to me some papers I would be appreciative. Thanks.

In my particular case, $L$ is like a distributional derivative, $A$ is a second order elliptic term and $B$ is a zero order non-monotone term. Also, $D(L) \subset H$ is a compact embedding.