The Ljapunov–Schmidt Reduction for Some Critical Problems

On Coron's problem for weakly coupled elliptic systems

“…In , the authors extended their results in higher dimension . In both papers it is assumed that (here denotes the first eigenvalue of with homogeneous Dirichlet boundary conditions in ), and this plays a crucial role: indeed, as remarked by Chen and Zou, system with bounded, and can be considered a critically coupled version of the Brezis–Nirenberg problem and it is well known that in any dimension admits a positive solution (for an arbitrary bounded domain) if and only if (see for this result, and the survey for a more extended discussion).…”

“…and it is well known that in any dimension N ≥ 4 (1.2) admits a positive solution (for an arbitrary bounded domain) if and only if −ν 1 (Ω) < λ < 0 (see [5] for this result, and the survey [22] for a more extended discussion).…”

Topics in elliptic problems: from semilinear equations to shape optimization

“…We recommend the survey [157] for more results in the critical case. Therein, the reader can also find a nice and simple general explanation of the use of the Lyapunov-Schmidt reduction method as a powerful and useful technique to build solutions to semilinear elliptic problems.…”

Singular limits in higher order Liouville-type equations

“…The first step is to obtain a priori estimates for the problem    Lε(φ) = h in Ωε, Bj φ = 0, |j| ≤ m − 1 on ∂Ωε, Ωε χiZij φ = 0 for all j = 0, ..., 2m, i = 1, ..., k which involves more orthogonality conditions than those in (4.1). Notice that in the case m = 1, independently from the nonlinearity, a key step in order to prove such result is the fact that the operator Lε satisfies maximum principle in Ωε outside large balls, see ( [23], Lemma 3.1) and [27] for the exponential-type nonlinearity or the surveys [49] and references therein for other nonlinearity issue such as the Brezis-Nirenberg Problem or the Coron's Problem. For our Lε this is not more true and we need a different approach.…”