On multiple solutions for nonlocal fractional problems via \(\nabla\)-theorems. (Q2402126)

Let \(s\in(0,1)\), and let \(n\in\mathbb{N}\), with \(n>2s\). Consider also an open bounded set \(\Omega\subset\mathbb{R}^n\) with continuous boundary. In this paper, the authors study the multiplicity of weak solutions to the following nonlocal elliptic problem \[ \begin{cases}\mathcal{L}_Ku+\lambda u+f(x,u)=0, \;\;\;&\text{in} \;\;\Omega,\\ u=0 &\text{in} \;\;\mathbb{R}^n\setminus\Omega.\end{cases}\eqno{(P)} \] Here, \(\lambda\in\mathbb{R}\) is a parameter, \(f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}\) si a Carathéodory function, and \(\mathcal{L}_K\) is the non-local operator defined by \[ \mathcal{L}_Ku(x)=\int_{\mathbb{R}^n}\left(u(x+y)-u(x-y)-2u(x)\right)K(y)dy, \;\;x\in\mathbb{R}^n. \] The function \(K:\mathbb{R}^n\setminus\{0\}\rightarrow(0,+\infty)\) involved in the definition of \(\mathcal{L}_K\) is assumed to satisfy the following conditions \[ \int_{\mathbb{R}^n}\min\{|x|^2,1\}K(x)dx<+\infty, \;\;\text{and} \;\;\inf_{x\in\mathbb{R}^n\setminus\{0\}}K(x)|x|^{n+2s}>0, \] and when \(K(x)=|x|^{-(n+2s)}\), \(\mathcal{L}_K\) is the well known fractional Laplacian operator \(-(-\Delta)^s\). Finally, the operator \(\mathcal{L}_K\) is defined in the space of all measurable functions \(g:\mathbb{R}^n\rightarrow\mathbb{R}\), such that \[ g_{\mid\Omega}\in L^2(\Omega), \;\;g_{\mid\mathbb{R}^n\setminus\Omega}=0,\;\;\text{and} \;\;\int_{\mathbb{R}^n\times\mathbb{R}^n}|g(x)-g(y)|^2K(x-y)dxdy<+\infty. \] After recalling the notion of weak solution for problem \((P)\), the authors prove that, if the nonlinearity \(f\) satisfies certain superlinear growth conditions, then, for each eigenvalue \(\lambda_k\) of the operator \(-\mathcal{L}_K\), there exists a left neighborhood \(\mathcal{O}_k\) of \(\lambda_k\) such that problem \((P)\) admits at least three nontrivial weak solutions, for all \(\lambda\in\mathcal{O}_k\). The authors use variational methods to prove their main result. In particular, the key tool is a critical point theorem established in \textit{A. Marino} and \textit{C. Saccon} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25, No. 3--4, 631--665 (1997; Zbl 1033.35026)].