Anuradha, V.; Dickens, S.; Shivaji, R.

Existence results for non-autonomous elliptic boundary value problems

Electron. J. Differ. Equ. 1994(4), (1994)


Summary: 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.

Mathematics Subject Classification



elliptic boundary value problems, semipositone