Multiplicity of positive solutions for a fractional \(p\& q\)-Laplacian problem in \(\mathbb{R}^N\) (Q2033275)

This paper is concerned with the following fractional \(p\&q\)-Laplacian problem: \[ \begin{aligned}\begin{cases} (-\Delta)^s_p u + (-\Delta)^s_qu + V(\varepsilon x) (|u|^{p-2}u +|u|^{q-2}u) = f(u) \quad & \text{in }\mathbb R^N,\\ u\in W^{s,p}(\mathbb R^N)\cap W^{s,q}(\mathbb R^N), u>0 & \text{in }\mathbb R^N, \end{cases}\end{aligned} \] where \(s\in (0, 1), \varepsilon> 0\) is a small parameter, \(2\leq p < q < N/s, (-\Delta)^s_t,\) with \(t \in\{p, q\}\), is the fractional \((s, t)\)-Laplacian operator, \(V: \mathbb R^N\rightarrow\mathbb R\) is a continuous function satisfying the global Rabinowitz condition \(V_{\infty} = \liminf_{|x|\rightarrow\infty}V(x) > V_0 = \inf_{x\in\mathbb R^N} V (x) >0,\) where both cases \(V_{\infty} <\infty\) and \(V_{\infty} =\infty\) are considered. The nonlinearity \(f: \mathbb R\rightarrow\mathbb R\) is a continuous function with subcritical growth. The main results are proved by variational methods. Especially, using the Nehari manifold method and Ljusternik-Schnirelmann category theory, the authors prove that the above problem admits multiple solutions for \(\varepsilon > 0\) small enough.