Abstract
We completely classify the behaviour near , as well as at when , of all positive solutions of in , where is a domain in () and . Here, and satisfy . Our classification depends on the position of relative to the critical exponent (with if ). We prove the following: if , then any positive solution has either (1) a removable singularity at , or (2) a weak singularity at (, where denotes the fundamental solution of the Laplacian), or (3) , where and are uniquely determined positive constants (a strong singularity). If (for ), then is a removable singularity for all positive solutions. Furthermore, for any positive solution in , we show that it is either constant or has a nonremovable singularity at (weak or strong). The latter case is possible only for , where we use a new iteration technique to prove that all positive solutions are radial, nonincreasing and converging to any nonnegative number at . This is in sharp contrast to the case of and , when all solutions decay to . Our classification theorems are accompanied by corresponding existence results in which we emphasise the more difficult case of , where new phenomena arise.
Citation
Joshua Ching. Florica Cîrstea. "Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term." Anal. PDE 8 (8) 1931 - 1962, 2015. https://doi.org/10.2140/apde.2015.8.1931
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