*Accepted Paper*

**Inserted:** 22 dec 2022

**Last Updated:** 9 feb 2024

**Journal:** Proc. Roy. Soc. Edinburgh Sec. A: Math

**Pages:** 34

**Year:** 2023

**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 $\alpha, \beta \in \mathbb{R}$ driven by the sum of two nonlocal operators:

$(-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha

u

^{p-2}u+\beta

u

^{q-2}u\;\;\text{in }\Omega, \quad u=0\;\;\text{in } \mathbb{R}^d \setminus \Omega, \ \ \ \qquad \quad \mathrm{(P)}
$

where $\Omega \subset \mathbb{R}^d$ is a bounded open set. Depending on the values of $\alpha,\beta$, we investigate the existence and non-existence of positive solutions to (P). A continuous curve in the two-dimensional $(\alpha,\beta)$-plane is constructed, which separates the regions of the existence and non-existence of positive solutions. We prove that the first Dirichlet eigenfunctions of the fractional $p$-Laplace and fractional $q$-Laplace operators are linearly independent and this plays an important role in the formation of the curve. Further, we demonstrated that every nonnegative solution of (P) is globally bounded.

**Keywords:**
Nehari manifold, generalized eigenvalue problems, fractional $(p,q)$-Laplace operator, positive solution, linear independence of the eigenfunctions

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