@article{Zhu2023, 
author = {Shuhai Zhu},
title = {Existence and multiplicity of solutions for a Schrödinger type equations involving the fractional    p  (  x  )-Laplacian},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {7},
pages = {16320-16339},
keywords = {fractional p(x)-Laplacian, fractional Sobolev space with variable exponent, variational method, fountain theorem},
url = {https://www.sciopen.com/article/10.3934/math.2023836},
doi = {10.3934/math.2023836},
abstract = {We are concerned with the following Schrödinger type equation with variable exponents     (  −      Δ          p      (      x      )            )          s        u  +  V  (  x  )      |    u            |              p      (      x      )      −      2        u  =  f  (  x  ,  u  )          in                    R              N        ,where    (  −      Δ          p      (      x      )            )          s       is the fractional    p  (  x  )-Laplace operator,    s  ∈  (  0  ,  1  ),    V  :            R              N        →  (  0  ,  +  ∞  ) is a continuous potential function, and    f  :            R              N        ×      R    →      R   satisfies the Carathéodory condition. We study the nonlinearity of this equation which is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. By using variational techniques and the fountain theorem, we obtain the existence and multiplicity of nontrivial solutions. Furthermore, we show that the problem has a sequence of solutions with high energies.}
}