Uniqueness of Positive Solutions for Elliptic Dirichlet Problems

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In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB, where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods ... continued below

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iii, 50 leaves : ill.

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Ali, Ismail, 1961- December 1990.

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  • Ali, Ismail, 1961-

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In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB,
where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order.
We also present a regularity result on linear elliptic equation where a coefficient has critical growth.

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iii, 50 leaves : ill.

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  • December 1990

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  • Aug. 22, 2014, 6 p.m.

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  • May 4, 2016, 1:37 p.m.

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Ali, Ismail, 1961-. Uniqueness of Positive Solutions for Elliptic Dirichlet Problems, dissertation, December 1990; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc330654/: accessed December 11, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .