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Nonlinear degenerate elliptic partial differential equations with critical growth conditions on the gradient

Authors: Kwon Cho and Hi Jun Choe
Journal: Proc. Amer. Math. Soc. 123 (1995), 3789-3796
MSC: Primary 35J70; Secondary 35J60
MathSciNet review: 1285981
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Abstract: We consider a nonlinear degenerate elliptic partial differential equation $ - {\operatorname{div}}(\vert\nabla u{\vert^{p - 2}}\nabla u) = H(x,u,\nabla u)$ with the critical growth condition on $ H(x,u,\nabla u) \leq g(x) + \vert\nabla u{\vert^p}$, where g is sufficiently integrable and p is between 1 and $ \infty $. Our first goal of this paper is to prove the existence of the solution in $ W_0^{1,p} \cap {L^\infty }$. The main idea is to obtain the uniform $ {L^\infty }$-estimate of suitable approximate solutions, employing a truncation technique and radially decreasing symmetrization techniques based on rearrangements. We also find an example of unbounded weak solution of $ - {\operatorname{div}}(\vert\nabla u{\vert^{p - 2}}\nabla u) = \vert\nabla u{\vert^p}$ for $ 1 < p \leq n$.

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Keywords: Existence, degenerate, critical growth condition, rearrangements, spherical symmetric unbounded solution
Article copyright: © Copyright 1995 American Mathematical Society

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