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

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