The existence of bounded
positive solutions of semilinear elliptic boundary value problems of the type
Lu
= λf(x,u), x ∈ Ω,
(1)
u(x)
= 0, x ∈ ∂Ω,
(2)
will be proved in unbounded domains Ω ⊂ Rn, n ≥ 2, with boundary
∂Ω ∈ C2+α,0 < α < 1, where λ is a positive constant and
(3)
Di= ∂∕∂xi, i = 1,…,n. The existence of a bounded positive solution of 1 in the
entire space Rn is proved also by the same procedure. The regularity and additional
hypotheses H1–H5 to be imposed on L and f are stated in §2. In particular, the
assumption f(x,0) = 0 for all x ∈ Ω implies that the boundary value problem 1, 2
always has the trivial solution.
![∑n
Lu = − Di[aij(x)Dju]+ m (x)u, x ∈ Ω,
i,j=1](a130x.png)
