Geodesic
Existence results for non-autonomous elliptic boundary value problems
Electronic journal of differential equations, Tome 1994 (1994) no. 4
We study solutions to the boundary value problems

$\eqalign{-\Delta u(x) = \lambda f(x, u);\quad \ x \in \Omega\cr u(x) + \alpha(x) {\partial u(x)\over \partial n} = 0;\quad \ x \in \partial \Omega}$

where

$\lambda \in (\lambda_{n}, \lambda_{n + 1})$

where $\lambda_{k}$ is the k-th eigenvalue of $-\Delta$ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for $\lambda$ near $\lambda_{1}$, and for $\lambda$ large when f is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for $\lambda$ small.
Classification : 35J65
Keywords: elliptic boundary value problems, semipositone 
@article{EJDE_1994__1994_4_a0,
     author = {Anuradha,  V. and Dickens,  S. and Shivaji,  R.},
     title = {Existence results for non-autonomous elliptic boundary value problems},
     journal = {Electronic journal of differential equations},
     year = {1994},
     volume = {1994},
     number = {4},
     zbl = {0809.35023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_1994__1994_4_a0/}
}
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Anuradha,  V.; Dickens,  S.; Shivaji,  R. Existence results for non-autonomous elliptic boundary value problems. Electronic journal of differential equations, Tome 1994 (1994) no. 4. http://geodesic.mathdoc.fr/item/EJDE_1994__1994_4_a0/