PDE and Linear operators Consider a linear PDE which can be seen as 
L[u] = 0, where L is a linear differential operator on u.
Is there any theory that tries to study the PDE by sutdying the kernel of L?

Yes. This is the functional-analytic formulation of the study of linear PDEs, in which a linear differential operator $L$ is viewed as a linear operator between two appropriate vector spaces. For example, $L$ is a differential operator of order $k$ and $u$ is assumed to live on some domain $U$, then one might naturally think of considering $L$ as an operator from $C^k(U)\to C(U)$. ($C^k(U)$ denotes the space of $k$-times continuously differentiable functions on $U$, $C(U)$ the space of continuous functions on $U$.) A solution $u$ to the linear PDE $Lu = 0$ can then be thought of as an element of $\ker L \subset C^k(U)$.
Unfortunately, this formulation of the problem with the spaces $C^k(U)$ and $C(U)$ can be very difficult to deal with. But the reason this approach works well is that we actually have a lot of leeway in choosing our vector spaces. This leads one to the notion of weak solution and vector spaces such as Sobolev space and spaces of distributions, AKA generalized functions. Typically finding weak solutions, then converting weak solutions to classical (i.e. continuously differentiable, or similarly regular) solutions is easier than searching for a classical solution from the get-go.
This is a main approach in modern PDE, and is also applicable to differential operators that are not quite linear. It has much interplay with the study of the variational formulation of a PDE and with Fourier analysis.
