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Let Ω be a bounded domain in R 2 with smooth boundary, we study the following anisotropic elliptic problem { − ∇ ( a ( x ) ∇ u ) + a ( x ) u = 0 in Ω , u > 0 in Ω , ∂ u ∂ ν = u p on ∂ Ω , where ν denotes the outer unit normal vector to ∂ Ω , p > 1 is a large exponent and a ( x ) is a positive smooth function. We construct solutions of this problem which exhibit the accumulation of arbitrarily many boundary peaks at any isolated local maximum point of a ( x ) on the boundary. [ABSTRACT FROM AUTHOR]

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed. [ABSTRACT FROM AUTHOR]