Fractional (p, q)-Schrödinger Equations with Critical and Supercritical Growth

Abstract

AbstractIn this paper, we complete the study started in Ambrosio and Rădulescu (J Math Pures Appl (9) 142:101–145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrödinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional (p, q)-Laplacian problems: $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p}u+(-\Delta )^{s}_{q}u + V(\varepsilon x) (u^{p-1} + u^{q-1})= f(u)+u^{q^{*}_{s}-1} \, \text{ in } \mathbb {R}^{N}, \\ u\in W^{s, p}(\mathbb {R}^{N})\cap W^{s,q}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$ ( - Δ ) p s u + ( - Δ ) q s u + V ( ε x ) ( u p - 1 + u q - 1 ) = f ( u ) + u q s ∗ - 1 in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 in R N , where $$\varepsilon >0$$ ε > 0 is a small parameter, $$s\in (0, 1)$$ s ∈ ( 0 , 1 ) , $$1<p<q<\frac{N}{s}$$ 1 < p < q < N s , $$q^{*}_{s}=\frac{Nq}{N-sq}$$ q s ∗ = Nq N - s q is the fractional critical Sobolev exponent, $$(-\Delta )^{s}_{r}$$ ( - Δ ) r s , with $$r\in \{p, q\}$$ r ∈ { p , q } , is the fractional r-Laplacian operator, $$V:\mathbb {R}^{N}\rightarrow \mathbb {R}$$ V : R N → R is a positive continuous potential such that $$\inf _{\partial \Lambda }V>\inf _{\Lambda } V$$ inf ∂ Λ V > inf Λ V for some bounded open set $$\Lambda \subset \mathbb {R}^{N}$$ Λ ⊂ R N , and $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik–Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when $$f(t)=t^{\gamma -1}+\mu t^{\tau -1}$$ f ( t ) = t γ - 1 + μ t τ - 1 , with $$1< p<q<\gamma<q^{*}_{s}<\tau $$ 1 < p < q < γ < q s ∗ < τ and $$\mu >0$$ μ > 0 sufficiently small, by combining a truncation argument with a Moser-type iteration.