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EXISTENCE OF WEAK SOLUTIONS TO A CLASS OF SCHRÖDINGER TYPE EQUATIONS INVOLVING THE FRACTIONAL p-LAPLACIAN IN ℝN

  • Kim, Jae-Myoung (Department of Mathematics Education Andong National University) ;
  • Kim, Yun-Ho (Department of Mathematics Education Sangmyung University) ;
  • Lee, Jongrak (Department of Mathematics, Ewha Womans University)
  • Received : 2018.11.20
  • Accepted : 2019.01.24
  • Published : 2019.11.01

Abstract

We are concerned with the following elliptic equations: $$(-{\Delta})^s_pu+V (x){\mid}u{\mid}^{p-2}u={\lambda}g(x,u){\text{ in }}{\mathbb{R}}^N$$, where $(-{\Delta})_p^s$ is the fractional p-Laplacian operator with 0 < s < 1 < p < $+{\infty}$, sp < N, the potential function $V:{\mathbb{R}}^N{\rightarrow}(0,{\infty})$ is a continuous potential function, and $g:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ satisfies a $Carath{\acute{e}}odory$ condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter ${\lambda}$ for which the problem admits at least one nontrivial weak solution when the nonlinearity g has the subcritical growth condition.

Acknowledgement

Supported by : National Research Foundation of Korea

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