Geodesic
On solvability of Dirichlet problem to semilinear Schr\"odinger equation with singular potential
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 25-45

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We prove the existence of a positive solution to the Dirichlet problem for the semilinear Schrödinger equation with acritical singular potential. We assume that the boundary $\partial\Omega$ is average concave at the origin. 
@article{ZNSL_2006_336_a1,
     author = {A. V. Demyanov and A. I. Nazarov},
     title = {On solvability of {Dirichlet} problem to semilinear {Schr\"odinger} equation with singular potential},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {25--45},
     publisher = {mathdoc},
     volume = {336},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a1/}
}
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A. V. Demyanov; A. I. Nazarov. On solvability of Dirichlet problem to semilinear Schr\"odinger equation with singular potential. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 25-45. http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a1/