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• W2806390631 abstract "Abstract We are concerned with the following singularly perturbed fractional Schrödinger equation: <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 /> <m:mtd columnalign=left> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi>ε</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msup> <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>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </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>u</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd /> <m:mtd columnalign=right> <m:mrow> <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:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd /> <m:mtd columnalign=left> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd /> <m:mtd columnalign=right> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>></m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>⁢</m:mo> <m:mtext> on </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:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> left{begin{aligned} &displaystyle{varepsilon^{2s}}{(-Delta)^{s}}u+V(x)u=% f(u)&&displaystyle{text{in }}{mathbb{R}^{N}}, &displaystyle uin{H^{s}}({mathbb{R}^{N}}),&&displaystyle u>0{text{ on }}{% mathbb{R}^{N}},end{aligned}right. where ε is a small positive parameter, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>N</m:mi> <m:mo>></m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:math> {N>2s} , and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <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>s</m:mi> </m:msup> </m:math> {{(-Delta)^{s}}} , with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {sin(0,1)} , is the fractional Laplacian. Using variational technique, we construct a family of positive solutions <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {{u_{varepsilon}}in{H^{s}}({mathbb{R}^{N}})} which concentrates around the local minima of V as <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>ε</m:mi> <m:mo>→</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {varepsilonto 0} under general conditions on f which we believe to be almost optimal." @default.
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• W2806390631 date "2018-05-29" @default.
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• W2806390631 title "Singularly Perturbed Fractional Schrödinger Equations with Critical Growth" @default.
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