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 |
.
Abstract
In this paper, we consider the nonhomogeneous
semilinear elliptic equation
where 0, 1 < p < (N+2)/(N-2), if N3, 1 < p < , if N = 2, h(x) H^{-1}(^{N} ), 0 h(x) 0 in ^{N}, K(x) is a positive, bounded and continuous function on ^{N}. We prove that if K(x) K_{} > 0 in ^{N}, and K( x) = K_{}, then there exists a positive constant such that (*)_{} has at least two solutions if (0,) and no solution if > . Furthermore, (*)_{} has a unique solution for = provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (*)_{} at = .
(*)_{} | - u + u = K(x)u^{p} + h(x) in ^{N}, u > 0 in ^{N} , u H ^{1}(^{N} ), |
where 0, 1 < p < (N+2)/(N-2), if N3, 1 < p < , if N = 2, h(x) H^{-1}(^{N} ), 0 h(x) 0 in ^{N}, K(x) is a positive, bounded and continuous function on ^{N}. We prove that if K(x) K_{} > 0 in ^{N}, and K( x) = K_{}, then there exists a positive constant such that (*)_{} has at least two solutions if (0,) and no solution if > . Furthermore, (*)_{} has a unique solution for = provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (*)_{} at = .
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