On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters

Abstract

For $s_1,s_2\in (0,1)$ and $p,q\in (1,\infty )$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\hat{I}\pm ,{\hat{I}}^2\in \mathbb{R}$ driven by the sum of two nonlocal operators: \begin{equation*} (-Δ)^{s_1}_p u+(-Δ)^{s_2}_q u=α|u|^{p-2}u+β|u|^{q-2}u\;\;\text{in }Ω, \quad u=0\;\;\text{in } \mathbb{R}^d \setminus Ω, \ \ \ \qquad \quad \mathrm{(P)} \end{equation*} where $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ is a bounded open set. Depending on the values of $\hat{I}\pm ,{\hat{I}}^2$, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional $(\hat{I}\pm ,{\hat{I}}^2)$-plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional $p$-Laplace and fractional $q$-Laplace operators are linearly independent, which plays an essential role in the formation of the curve. Furthermore, we establish that every nonnegative solution of (P) is globally bounded.

35 pages, 2 figures (In the latest version, we have revised the proof of Theorem 1.6-(i) by constructing a suitable test function to demonstrate that the solution is nonzero.)