Critical Perturbation Results for a Mixed Boundary Value Problem

Abstract

Let K be a given positive function on a bounded domain Ω of ℝⁿ, n ≥ 3. The authors consider a nonlinear variational problem of the form: $-\Delta u=K\vert u\vert^{4\over{n-2}}u$ in Ω with mixed Dirichlet-Neumann boundary conditions. It is a non-compact variational problem, in the sense that the associated energy functional J fails to satisfy the Palais-Smale condition. This generates concentration and blow-up phenomena. By studying the behaviors of non-precompact flow lines of a decreasing pseudogradient of J, they characterize the points where blow-up phenomena occur, the so-called critical points at infinity. Such a characterization combined with tools of Morse theory, algebraic topology and dynamical system, allow them to prove critical perturbation results under geometrical hypothesis on the boundary part in which the Neumann condition is prescribed.

Keywords

Critical elliptic equations / Variational methods / Asymptotic analyzes / Critical points at infinity / 35J20

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