On Least Energy Solutions to a Pure Neumann Lane–Emden System: Convergence, Symmetry Breaking, and Multiplicity

We consider the following Lane–Emden system with Neumann boundary conditions: − Δ u = | v | q − 1 v in Ω , − Δ v = | u | p − 1 u in Ω , ∂ ν u = ∂ ν v = 0 on ∂ Ω , where Ω is a bounded smooth domain of R N with N ≥ 1 . We study the multiplicity of solutions and the convergence of least energy (nodal) solutions (l.e.s.) as the exponents p , q > 0 vary in the subcritical regime 1 / ( p + 1 ) + 1 / ( q + 1 ) > ( N − 2 ) / N , or in the critical case 1 / ( p + 1 ) + 1 / ( q + 1 ) = ( N − 2 ) / N with some additional assumptions. We consider, for the first time in this setting, the cases where one or two exponents tend to zero, proving that l.e.s. converge to a problem with a sign nonlinearity. Our approach is based on an alternative characterization of the least energy levels in terms of the nonlinear eigenvalue problem <mml:mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2e