DOI: 10.1017/s000497272610207x ISSN: 0004-9727

SOME RESULTS ON INDEFINITE SUBLINEAR ELLIPTIC RADIAL PROBLEMS WITH SIMULTANEOUS DIRICHLET AND NEUMANN BOUNDARY CONDITIONS

URIEL KAUFMANN
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Abstract

Let B be a ball in

R N $\mathbb {R}^{N}$ double struck upper R Superscript upper N
,
N ≥ 1 $N\geq 1$ upper N greater than or equals 1
, let
m ∈ C ( B ― ) $m\in C(\overline {B}) $ m element of upper C left parenthesis upper B overbar right parenthesis
be a radial function that changes sign in B and let
0 < q < 1 $0<q<1$ 0 less than q less than 1
. We consider the existence and nonexistence of (nontrivial) nonnegative solutions and positive solutions for semilinear elliptic problems of the form
− Δ u = m ( x ) u q $-\Delta u=m( x) u^{q}$ minus normal upper Delta u equals m left parenthesis x right parenthesis u Superscript q
in B ,
u = ∂ ν u = 0 $u=\partial _{\nu }u=0$ u equals partial differential Subscript nu Baseline u equals 0
on
∂ B $\partial B$ partial differential upper B
, where
∂ ν $\partial _{\nu }$ partial differential Subscript nu Baseline
is the exterior normal derivative.

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