Multiscale Weak Compactness in Metric Spaces

On the existence of multiple solutions for fractional Brezis–Nirenberg‐type equations

“…Remark In Step 3 above, to show E satisfies (PS) c condition for any $$c<c^*$$, we proceed in a classical way. But to recover compactness, that is, to prove (PS) c condition, one may use profile decomposition theorem [13, 15, 16, 20, 23, 24, 31, 33, 34]. The main steps are the following: (1)Consider a (PS) c sequence $$\lbrace u_k\rbrace _{k \ge 1}$$ for E , that is, $$E(u_k) \rightarrow c$$ and $$E^{\prime }(u_k) \rightarrow 0$$ in $$(X_0(\Omega ))^{\prime }$$ as $$k \rightarrow \infty$$, thereby showing that sequence $$\lbrace u_k\rbrace _{k \ge 1}$$ is bounded in $$X_0(\Omega )$$. (2)The weak limit of $$\lbrace u_k\rbrace _{k \ge 1}$$ (up to a subsequence), u (0) solves $$(\mathcal {P})$$. (3)If the convergence of …”

“…In Step 3 above, to show 𝐸 satisfies (PS) 𝑐 condition for any 𝑐 < 𝑐 * , we proceed in a classical way. But to recover compactness, that is, to prove (PS) 𝑐 condition, one may use profile decomposition theorem[13,15,16,20,23,24,31,33,34].…”

On nonlinear Schrödinger equations on the hyperbolic space

“…), ϕ = 0, for every ϕ ∈ H 1,2 SO(N ) (B N ) and λ ∈ (0, λ (ω)). Hence, J λ (u Profile decomposition methods can be useful in order to study similar problems when a lack of compactness occurs (see, among others, the recent papers [15,16]). A further and more general investigation of this topics will be included in the forthcoming book [37].…”

On nonlinear Schrödinger equations on the hyperbolic space

“…Remark 4.2. Profile decomposition methods can be useful in order to study similar problems when a lack of compactness occurs (see, among others, the recent papers [15,16]). A further and more general investigation of this topics will be included in the forthcoming book [37].…”