Multiple normalized solutions for fractional elliptic problems
Thin Van Nguyen, Vicenţiu D. Rădulescu
Abstract
Abstract In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional p -Laplace problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:msubsup> <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:mi>p</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo></m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi mathvariant="script">𝒱</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>ξ</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>v</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:mi>v</m:mi> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>v</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo separator="true"> </m:mo> <m:mrow> <m:mtext>in </m:mtext> <m:mo></m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:msub> <m:mrow> <m:mpadded width="+1.7pt"> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>v</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mpadded> <m:mo></m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:msup> <m:mi>a</m:mi> <m:mi>p</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \left\{\begin{aligned} \displaystyle{}(-\Delta)_{p}^{s}v+\mathcal{V}(\xi x)% \lvert v\rvert^