Weakly Nonlinear Dirichlet Problems On Long Or Thin Domains

Weakly Nonlinear Dirichlet Problems on Long or Thin Domains

In this paper, we discuss the existence, uniqueness and asymptotic behavior of positive solutions of the equation −[capital Greek]Delta[italic]u = [lowercase Greek]Lambda[function]ƒ([italic]u) in [capital Greek]Omega[surmounted by macron] [times symbol] [−[italic]n, [italic]n], [and] [italic]u = 0 on [partial derivative/boundary/degree of a polynomial symbol]([capital Greek]Omega[surmounted by macron] [times symbol] [−[italic]n, [italic]n]) for [italic]n large. Here [capital Greek]Omega[surmounted by macron] is a bounded domain in [italic capital]R[superscript italic]k with smooth boundary. Note that by rescaling the equation (including [lowercase Greek]Lambda), our theory covers problems on domains ([set membership symbol][capital Greek]Omega[surmounted by macron]) [times symbol] [−1,1] where [set membership symbol] is small.

Weakly Nonlinear Dirichlet Problems on Long Or Thin Domains

Presents a basic theory for nonlinear elliptic equations on long or thin domains for Dirichlet boundary conditions. This book describes Dirichlet problems which are of significant interest in applications.

Weakly Nonlinear Dirichlet Problems on Long Or Thin Domains