| Popis: |
$$-Delta u(x) = lambda f(x, u);quad x in Omega$$ $$u(x) + alpha(x) frac{partial u(x)}{partial n} = 0;quad x in partial Omega$$ where $lambda > 0$, $Omega$ is a bounded region in $Bbb{R}^N$; $N geq 1$ 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. |
