Multiplicity and concentration of solutions to fractional anisotropic Schrödinger equations with exponential growth

Abstract In this paper, we consider the Schrödinger equation involving fractional $$(p,p_1,\dots ,p_m)$$ ( p , 1 ⋯ m ) -Laplacian as follows $$\begin{aligned} (-\Delta )_{p}^{s}u+\sum _{i=1}^{m}(-\Delta )_{p_i}^{s}u+V(\varepsilon x)(|u|^{(N-2s)/2s}u+\sum _{i=1}^{m}|u|^{p_i-2}u)=f(u)\;\text{ in }\; {\mathbb {R}}^{N}, \end{aligned}$$ - Δ s u + ∑ i = V ε x | N 2 / f in R where $$\varepsilon $$ is a positive parameter, $$N=ps, s\in (0,1), 2\le p<p_1< \dots< p_m<+\infty , m\ge 1$$ ∈ 0 ≤ < ∞ ≥ . The nonlinear function f has exponential growth and potential V continuous satisfying some suitable conditions. Using penalization method Ljusternik–Schnirelmann theory, study existence, multiplicity concentration of nontrivial nonnegative solutions for small values parameter. our best knowledge, it first time that above problem studied.