Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems

Abstract

In this paper we consider the following problemwhere λ > 0 is a parameter. We assume lim |x|→∞ f (x, u) =f (u) uniformly on any compact subset of [0, ∞), and do not require f (x, u) f (u) for all x ∈ R N . We prove that there exists +∞ > λ * > 0 such that ( ) has exactly two positive solutions for λ ∈ (0, λ * ), no solution for λ > λ * , a unique solution for λ = λ * , (λ * , u * ) is a turning point in C 2,α (R N ) ∩ W 2,2 (R N ), and further analyses of the set of positive solutions are made.

A supercritical scalar field equation with a forcing term

“…Remark 1.1 We give some comments on assumption (1.5). [1,5,9,10,12,21]). The proof of Theorem 1.1 is based on the construction of approximate solutions and the supersolution-subsolution method.…”

Existence and multiplicity of positive solutions for a class of nonlinear elliptic problems

“…where g satisfies some suitable conditions and f ∈ H −1 (R N )\{0} is nonnegative, has been the focus of a great deal of research by several authors [1,6,7,12] and the existence of at least two positive solutions was proved.…”

On a boundary value problem in the half-space

“…The Laplace equation in the whole space R N has been studied extensively, see, for example, [2][3][4][5][6][7][8][9]. In recent years the existence and multiplicity of the Laplace equation in the half-space R N + have been gained much interest, see [10][11][12]14].…”