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

2008; Wiley; Volume: 77; Issue: 2 Linguagem: Inglês

10.1112/jlms/jdm104

1469-7750

Catherine Bandle, Vitaly Moroz, Wolfgang Reichel,

Advanced Mathematical Modeling in Engineering

Abstract Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential μ/δ( x , ∂Ω) 2 , where δ( x , ∂Ω) 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].