Geodesic
Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity
Matematičeskie trudy, Tome 19 (2016) no. 1, pp. 91-105

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We consider the question of the existence of the Dirichlet problem for second-order elliptic equations with spectral parameter and a nonlinearity discontinuous with respect to the phase variable. Here it is not assumed that the differential operator is formally selfadjoint. Using the method of upper and lower solutions, we establish results on the existence of nontrivial (positive and negative) solutions under positive values of the spectral parameter for the problems under study. 
@article{MT_2016_19_1_a3,
     author = {V. N. Pavlenko and D. K. Potapov},
     title = {Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity},
     journal = {Matemati\v{c}eskie trudy},
     pages = {91--105},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2016_19_1_a3/}
}
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V. N. Pavlenko; D. K. Potapov. Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity. Matematičeskie trudy, Tome 19 (2016) no. 1, pp. 91-105. http://geodesic.mathdoc.fr/item/MT_2016_19_1_a3/