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

Abstract

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