Fractional eigenvalue problems on $\mathbb{R}^N$

Abstract

Let $N\geq 2$ be an integer. For each real number $s\in(0,1)$ we denote by $(-\Delta)^s$ the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem $(-\Delta)^s u=\lambda V(x)u$ on $\mathbb{R}^N$, where $V:\mathbb{R}^N\rightarrow\mathbb{R}$ is a given function. Under suitable conditions imposed on $V$ we show the existence of an unbounded, increasing sequence of positive eigenvalues. Next, we perturb the above eigenvalue problem with a fractional $(t,p)$-Laplace operator, when $t\in(0,1)$ and $p\in(1,\infty)$ are such that $t<s$ and $s-N/2=t-N/p$. We show that when the function $V$ is nonnegative on $\mathbb{R}^N$, the set of eigenvalues of the perturbed eigenvalue problem is exactly the unbounded interval $(\lambda_1,\infty)$, where $\lambda_1$ stands for the first eigenvalue of the initial eigenvalue problem.

Keywords

• fractional laplacian
• eigenvalue problem
• weak solution
• minimization problem
• nehari manifold