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

• Autores: Catherine Bandle, Vitaly Moroz, Wolfgang Reichel
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 77, Nº 2, 2008, págs. 503-523
• Idioma: inglés
• DOI: 10.1112/jlms/jdm104
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• Resumen

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