Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
Abstract In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>ϕ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>H</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd columnalign="left"> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>ϕ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd columnalign="left"> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd columnalign="left"> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ u{ >}0,\quad \hfill & \text{in} {\mathbb
Authors
- Zhipeng Yang (ORCID: https://orcid.org/0000-0002-5454-3778)
- Changyang Mu
- Wei Zhang (ORCID: https://orcid.org/0000-0002-7911-0514)
Institutions
- Yunnan Normal University (CN)
- Yunnan University of Finance And Economics (CN)
Publication Details
- Journal
- Advanced Nonlinear Studies
- Published
- 2024-09-20
- DOI
- https://doi.org/10.1515/ans-2023-0147
- Citations
- 1
- Primary Topic
- Nonlinear Partial Differential Equations
- Type
- article
- Field-Weighted Citation Impact
- 0.97