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Semilinear elliptic problems with nonlinearities depending on the derivative

David Arcoya, Naira del Toro (2003)

Commentationes Mathematicae Universitatis Carolinae

We deal with the boundary value problem - Δ u ( x ) = λ 1 u ( x ) + g ( u ( x ) ) + h ( x ) , x Ω u ( x ) = 0 , x Ω where Ω N is an smooth bounded domain, λ 1 is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on Ω , h L max { 2 , N / 2 } ( Ω ) and g : N is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that g satisfies certain conditions on the origin and at infinity.

Smooth bifurcation for a Signorini problem on a rectangle

Jan Eisner, Milan Kučera, Lutz Recke (2012)

Mathematica Bohemica

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof...

Spatial patterns for reaction-diffusion systems with conditions described by inclusions

Jan Eisner, Milan Kučera (1997)

Applications of Mathematics

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.

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