Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions

TL;DR

This paper investigates the existence and boundedness of solutions to a critical fractional Laplacian problem with mixed boundary conditions, revealing multiple solutions in the sublinear case and at least one in the superlinear case.

Contribution

It establishes new existence results for solutions to a critical fractional problem with mixed boundary conditions, including multiple solutions in the sublinear case.

Findings

Multiple solutions for sublinear nonlinearities when 0<q<1.

Existence of at least one solution for superlinear case 1<q<2_s^*-1.

Solutions are proven to be bounded.

Abstract

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} (-\Delta)^su=\lambda u^q+u^{2_s^*-1},\ u>0\quad\text{in }\Omega,\\[3pt] \mkern+51mu u=0\quad\text{on } \Sigma_{\mathcal{D}}\\ \mkern+36mu \displaystyle \frac{\partial u}{\partial \nu}=0\quad\text{on } \Sigma_{\mathcal{N}} \end{array} \right. \end{equation*} where $\Omega\subset\mathbb{R}^N$ 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, $\lambda>0$, $\nu$ is the outwards normal to $\partial\Omega$, $\Sigma_{\mathcal{D}}$, $\Sigma_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partial\Omega$ such that $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$, and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$-dimensional submanifold of $\partial\Omega$.\newline In particular, we will prove that, for the sublinear case $0<q<1$, there exists at least two solutions for every $0<\lambda<\Lambda$ for certain $\Lambda\in\mathbb{R}$ while, for the superlinear case $1<q<2_s^*-1$, we will prove that there exists at least one solution for every $\lambda>0$. We will also prove that solutions are bounded.