## 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

### Keywords:

method of sub-super solutions; multiple solutions
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