Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian
Let \(\Omega\subset \mathbb{R}^n\) (\(n\geq 2\)) be a bounded smooth domain and let \(s\in (0,1)\). In this paper, the authors deal with the existence of nodal and multiple solutions to the problem \[ (-\Delta)^su=f(x,u) \text{in } \Omega, \quad u=0 \text{ on } \partial \Omega,\tag{P} \] where \((-\Delta)^s\) is the fractional Laplacian and \(f\in C(\overline{\Omega}\times \mathbb{R})\) is a function which satisfies the following subcritical and Ambrosetti-Rabinowitz growth conditions at infinity {\parindent=7mm\begin{itemize}\item[1)] \(\sup_{(x,t)\in\overline{\Omega}\times \mathbb{R} }\frac{|f(x,t)|}{1+|t|^{p-1}}<+\infty\) for some \(p\in (2,\frac{2n}{n-2s})\) \item[2)] there exist \(\mu>2\) and \(M>0\) such that: \[ 0<\int_0^t f(x,s)ds\leq \frac{1}{\mu}f(x,t) \text{ for all } x\in \overline{\Omega} \text{ and } t\in \mathbb{R}\setminus (-M,M). \] \end{itemize}} By generalizing the method of invariant sets of descending flow to the case of elliptic equations involving the fractional Laplacian and using the well-known \(s\)-harmonic extension introduced by \textit{L. Caffarelli} and \textit{L. Silvestre} [Commun. Partial Differ. Equations 32, No. 8, 1245--1260 (2007; Zbl 1143.26002)] to transform the nonlocal problem \((P)\) in \(\Omega\) to a local problem in the cylinder \(\Omega \times (0,+\infty)\), the authors obtain the following results: Assume that \(\lim_{t\rightarrow 0}f(x,t)/t=0\) uniformly in \(x\in \overline{\Omega}\). Then, the problem \((P)\) admits a positive solution, a negative solution, and a nodal solution. Moreover, if in addition the function \(f(x,t)/|t|\) is nondecreasing in \(t\in \mathbb{R}\) for every \(x\in \overline{\Omega}\), then there exists a least energy nodal solution with exactly two nodal domains. Assume that \(f(x,t)=f(x,-t)\) for all \(x\in \overline{\Omega}\) and all \(t\in \mathbb{R}\). Then, the problem \((P)\) admits infinitely many nodal solutions. Moreover, if in addition the function \(f(x,t)/|t|\) is nondecreasing in \(t\in \mathbb{R}\) for every \(x\in \overline{\Omega}\), each of these nodal solutions has at most \(m+1\) nodal domains.
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