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Existence results for non-autonomous elliptic boundary value problems. (English) Zbl 0809.35023

Summary: We study solutions to the boundary value problems \[ -\Delta u(x)= \lambda f(x,u);\quad x\in \Omega, \qquad u(x)+ \alpha(x) {{\partial u(x)} \over {\partial n}}=0; \quad x\in \partial \Omega \] where \(\lambda>0\), \(\Omega\) is a bounded region in \(\mathbb{R}^ N\); with smooth boundary \(\partial\Omega\), \(\alpha(x) \geq 0\), \(n\) is the outward unit normal, and \(f\) is a smooth function such that it has either sublinear or restricted linear growth in \(u\) at infinity, uniformly in \(x\).
We also consider \(f\) such that \(f(x,u)u \leq 0\) uniformly in \(x\), when \(| u|\) is large. Without requiring any sign condition on \(f(x,0)\), thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given \(\lambda\in (\lambda_ n, \lambda_{n+1})\) 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.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
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