Displaying similar documents to “Semilinear elliptic problems with nonlinearities depending on the derivative”

Bifurcations for a problem with jumping nonlinearities

Lucie Kárná, Milan Kučera (2002)

Mathematica Bohemica

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A bifurcation problem for the equation Δ u + λ u - α u + + β u - + g ( λ , u ) = 0 in a bounded domain in N with mixed boundary conditions, given nonnegative functions α , β ∈ L ∞ and a small perturbation g is considered. The existence of a global bifurcation between two given simple eigenvalues λ ( 1 ) , λ ( 2 ) of the Laplacian is proved under some assumptions about the supports of the functions α , β . These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to λ ( 1 ) , λ ( 2 ) .

Continuum spectrum for the linearized extremal eigenvalue problem with boundary reactions

Futoshi Takahashi (2014)

Mathematica Bohemica

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We study the semilinear problem with the boundary reaction - Δ u + u = 0 in Ω , ∂ u ∂ ν = λ f ( u ) on ∂ Ω , where Ω ⊂ ℝ N , N ≥ 2 , is a smooth bounded domain, f : [ 0 , ∞ ) → ( 0 , ∞ ) is a smooth, strictly positive, convex, increasing function which is superlinear at ∞ , and λ > 0 is a parameter. It is known that there exists an extremal parameter λ * > 0 such that a classical minimal solution exists for λ < λ * , and there is no solution for λ > λ * . Moreover, there is a unique weak solution u * corresponding to the parameter λ = λ * . In this paper, we continue to study the spectral properties...