Geodesic
On some noncoercive nonlinear equations
Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 323-332

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In this paper nonlinear equations $A(u)=h$ are studied, where the operator does not necessarily satisfy the well-known condition of coerciveness. With the equation $A(u)=h$, which is in general not solvable for an arbitrary right side $h$, we associate a certain equation of the form $B^*A(u)=h$, which is always solvable. Then the original equation $A(u)=h$ is solvable up to $\operatorname{Ker}B^*$. This gives a description of the domain of values of the original operator $A(u)$. Bibliography: 9 titles. 
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     title = {On some noncoercive nonlinear equations},
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Yu. A. Dubinskii. On some noncoercive nonlinear equations. Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 323-332. http://geodesic.mathdoc.fr/item/SM_1972_16_3_a0/