Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations
Eduardo Colorado, Giovanni Monica Bisci, Alejandro Ortega, Luca Vilasi
TL;DR
This work addresses a nonlocal critical problem driven by the spectral fractional Laplacian $(-\Delta)^s$ with mixed Dirichlet-Neumann boundary data and the critical exponent $2_s^*$. The authors develop a variational framework and apply a $\nabla$-theorem to obtain multiplicity of solutions as $\lambda$ approaches eigenvalues $\Lambda_k$ of the mixed-boundary spectral problem. Central contributions include a spectral decomposition of the energy space, an explicit energy threshold $\varepsilon_\lambda$ below which only the trivial critical point occurs, a PS threshold $c^*$, compactness results for PS sequences, and the verification of the $\nabla$-condition on a finite-codimension subspace; these yield at least two critical points for $\lambda$ in $(\Lambda_k-\delta_k, \Lambda_k)$. The results extend subcritical near-resonance multiplicity results to the critical nonlocal setting and provide explicit eigenvalue-neighborhood multiplicity near each $\Lambda_k$ in the mixed boundary context.
Abstract
We consider the following nonlocal critical problem with mixed Dirichlet-Neumann boundary conditions, \begin{equation} \left\{ \begin{array}{ll} (-Δ)^su=λu+|u|^{2_s^*-2}u &\text{in}\ Ω,\\ \mkern+38.5mu u=0& \text{on}\ Σ_{\mathcal{D}},\\ \mkern+24mu \displaystyle \frac{\partial u}{\partial ν}=0 &\text{on}\ Σ_{\mathcal{N}}, \end{array} \right. \end{equation} where $(-Δ)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $Ω\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $2_s^*=\frac{2N}{N-2s}$ denotes the critical fractional Sobolev exponent, $λ>0$ is a real parameter, $ν$ is the outwards normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$--dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$ and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$--dimensional submanifold of $\partialΩ$. By employing a $\nabla$-theorem we prove the existence of multiple solutions when the parameter $λ$ is in a left neighborhood of a given eigenvalue of $(-Δ)^s$.