Bull. Austral. Math. Soc. 72(3) pp.349--370, 2005.

Bifurcation of positive entire solutions for a semilinear elliptic equation

Tsing-San Hsu

Huei-Li Lin
Received: 25th April, 2005

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Abstract

In this paper, we consider the nonhomogeneous semilinear elliptic equation
(*)$\scriptstyle \lambda $ - $\displaystyle \Delta $u + u = $\displaystyle \lambda $K(x)up + h(x) in $\displaystyle \mathbb {R}$N, u > 0 in $\displaystyle \mathbb {R}$N  , u $\displaystyle \in $ H 1($\displaystyle \mathbb {R}$N ),

where $ \lambda $ $ \geq $ 0, 1 < p < (N+2)/(N-2), if N$ \ge $3, 1 < p < $ \infty $, if N = 2, h(x) $ \in $ H-1($ \mathbb {R}$N ), 0 $ \not \equiv $h(x) $ \geq $ 0 in $ \mathbb {R}$N, K(x) is a positive, bounded and continuous function on $ \mathbb {R}$N. We prove that if K(x) $ \geq $ K$\scriptstyle \infty $ > 0 in $ \mathbb {R}$N, and $ \lim \limits _{{\vert x\vert \rightarrow \infty }}^{}$K( x) = K$\scriptstyle \infty $, then there exists a positive constant $ \lambda ^{*}_{}$ such that (*)$\scriptstyle \lambda $ has at least two solutions if $ \lambda $ $ \in $ (0,$ \lambda ^{*}_{}$) and no solution if $ \lambda $ > $ \lambda ^{*}_{}$. Furthermore, (*)$\scriptstyle \lambda $ has a unique solution for $ \lambda $ = $ \lambda ^{*}_{}$ provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (*)$\scriptstyle \lambda $ at $ \lambda $ = $ \lambda ^{*}_{}$.

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(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199637 Z'blatt-MATH: 1097.35051

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