论文标题
凹形 - 凸照的关键问题对于具有混合边界条件的光谱分数laplacian
Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions
论文作者
论文摘要
在这项工作中，我们研究了解决以下关键分数问题的解决方案的存在，cove-convex非线性，\ begin {equination*} \ left \ {\ stray {array} {array} {l} {l}^su =λu=λu=λ \ mkern+51mu u = 0 \ quad \ text {on}σ_ {\ Mathcal {d}}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ displayStyle \ frac {\ partial U} \ end {array} \ right。 \ end {equation*} 其中$ω\ subset \ mathbb {r}^n $是一个平稳的界面，$ \ frac {1} {2} {2} <s <1 $，$ 0 <q <q <q <q <q <q <q <q <q <2_s^* - 1 $，$ q \ neq 1 $，$ 2_s^= $ 2_s^*= \ frac = \ frac {2n} $ so， $ν$是$ \ \partialΩ$，$σ_{\ Mathcal {d}} $，$σ_{\ Mathcal {n}} $是平稳的$（n-1）$ - dimenmional submanifolds of $ \partialΩ$，这样美元$σ_ {\ Mathcal {d}} \ cap \ cap \overlineς_ {\ Mathcal {n}} =γ$是平稳的$（n-2）$ - $ \partialΩ$。 $ 0 <λ<λ$对于某些$λ\ in \ mathbb {r} $，对于超线案例$ 1 <q <q <2_s^* - 1 $，我们将证明至少每$λ> 0 $都有至少一个解决方案。我们还将证明解决方案是有限的。
In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^su=λu^q+u^{2_s^*-1},\ u>0\quad\text{in }Ω,\\[3pt] \mkern+51mu u=0\quad\text{on } Σ_{\mathcal{D}}\\ \mkern+36mu \displaystyle \frac{\partial u}{\partial ν}=0\quad\text{on } Σ_{\mathcal{N}} \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $\frac{1}{2}<s<1$, $0<q<2_s^*-1$, $q\neq 1$, being $2_s^*=\frac{2N}{N-2s}$ the critical fractional Sobolev exponent, $λ>0$, $ν$ 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 $\partialΩ$.\newline In particular, we will prove that, for the sublinear case $0<q<1$, there exists at least two solutions for every $0<λ<Λ$ for certain $Λ\in\mathbb{R}$ while, for the superlinear case $1<q<2_s^*-1$, we will prove that there exists at least one solution for every $λ>0$. We will also prove that solutions are bounded.
