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Journal of the London Mathematical Society Advance Access originally published online on February 25, 2008
Journal of the London Mathematical Society 2008 77(2):503-523; doi:10.1112/jlms/jdm104
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© 2008 London Mathematical Society

‘Boundary blowup’ type sub-solutions to semilinear elliptic equations with Hardy potential

Catherine Bandle

Mathematisches Institut
Universität Basel
Rheinsprung 21
CH-4051 Basel
Switzerland
C.Bandle@gmx.ch

Vitaly Moroz

Department of Mathematics
Swansea University
Singleton Park
Swansea SA2 8PP
Wales
United Kingdom

Wolfgang Reichel

Institut für Analysis
Fakultät für Mathematik
Universität Karlsruhe
D-76128 Karlsruhe
Germany
Wolfgang.Reichel@math.uni-karlsruhe.de

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential µ/{delta}(x, {partial}{Omega})2, where {delta}(x, {partial}{Omega}) denotes the distance function. The size of this potential affects the existence of a certain type of solutions (large solutions): if µ is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen–Lindelöf type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller–Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived by Marcus, Mizel and Pinchover [Trans. Amer. Math. Soc. 350 (1998) 3237–3255].

2000 Mathematics Subject Classification 35J60 (primary), 35J70, 31B25 (secondary).

Received May 5, 2007; published online February 25, 2008.


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