If $L$ is an elliptic operator, can we say anything about the dimension of the Kernel of $L$? If so, is there any technique that resembles the "variation of parameters" for ODEs to solve an equation of the form $$ Lu=f\qquad\text{in }\mathbb{R}^N? $$ I imagine that the answer in general is negative. However, is there any known condition on $L$ and $f$ such that the answer is affirmative? Are there any references treating this argument?
Any help is very appreciated.