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 $V$ though. 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.

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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.