On the multiplicity and concentration for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" id="d1e19" altimg="si16.gif"><mml:mi>p</mml:mi></mml:math>-fractional Schrödinger equations

In this note we improve the recent results established in Ambrosio and Isernia (2018) concerning the multiplicity and concentration of positive solutions for the following class of p -fractional Schrödinger equations: ε s p ( − Δ ) p s u + V ( x ) | u | p − 2 u = f ( u ) + γ | u | p s ∗ − 2 u in R N where ε > 0 , s ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) , N > s p , γ ∈ { 0 , 1 } , p s ∗ = N p N − s p is the fractional critical exponent, ( − Δ ) p s is the fractional p -Laplacian operator, V is a continuous positive potential satisfying the Rabinowitz condition (Rabinowitz, 1992), and f is a superlinear continuous function with subcritical growth at infinity. Here we do not assume the well-known Ambrosetti–Rabinowitz condition (Ambrosetti and Rabinowitz, 1973) on f and we complete the study in the critical case γ = 1 .