Geodesic
Existence theorems for resonance boundary-value problems of elliptic type with discontinuous unbounded
Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 157-176

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The existence of a solution of the Dirichlet problem for a second order elliptic equation with non-linear part discontinuous in the phase variable is proved in the cases of resonance on the left and resonance on the right of the first eigenvalue of the differential operator in the situation where the Landesman–Lazer conditions do not hold.
Keywords: resonance elliptic boundary-value problems, discontinuous non-linearities, Landesman–Lazer condition. 
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     author = {E. A. Rozhdestvenskaya},
     title = {Existence theorems for resonance boundary-value problems of elliptic type with discontinuous unbounded},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a5/}
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E. A. Rozhdestvenskaya. Existence theorems for resonance boundary-value problems of elliptic type with discontinuous unbounded. Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 157-176. http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a5/