# ON DISCONTINUOUS ELLIPTIC PROBLEMS INVOLVING THE FRACTIONAL p-LAPLACIAN IN ℝN

• Kim, In Hyoun (Department of Mathematics Incheon National University) ;
• Kim, Yun-Ho (Department of Mathematics Education Sangmyung University) ;
• Park, Kisoeb (Department of Mathematics Incheon National University)
• 투고 : 2017.12.31
• 심사 : 2018.05.23
• 발행 : 2018.11.30

#### 초록

We are concerned with the following fractional p-Laplacian inclusion: $$(-{\Delta})^s_pu+V(x){\mid}u{\mid}^{p-2}u{\in}{\lambda}[{\underline{f}}(x,u(x)),\;{\bar{f}}(s,u(x))]$$ in ${\mathbb{R}}^N$, where $(-{\Delta})^s_p$ is the fractional p-Laplacian operator, 0 < s < 1 < p < $+{\infty}$, sp < N, and $f:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity f admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-{\Delta})^s_pu$ is replaced by any non-local integro-differential operator.

#### 과제정보

연구 과제 주관 기관 : Incheon National University

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