Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method∗∗∗. (7th April 2014)

• Record Type: Journal Article
• Authors: Figueiredo, Giovany M.; Santos, João R.
• Is Part Of: ESAIM. Volume 20:Number 2(2014:Apr.)
• Page Start: 389
• Page End: 415
• Publication Date: 2014-04-07
• Subjects: System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013068
• Languages: English
• ISSNs: 1292-8119

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem</p> <p> <disp-formula> <alternatives> <tex-math id="tex_eq1"><![CDATA[$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3}, \\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3}, \\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$]]></tex-math> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjx9jp0" xmlns:xlink="http://www.w3.org/1999/xlink" /> <mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo mathvariant="normal">(</mml:mo> <mml:msub> <mml:mi mathvariant="italic">P</mml:mi> <mml:mi mathvariant="italic">ε</mml:mi> </mml:msub> <mml:mo mathvariant="normal">)</mml:mo> <mml:mfenced close="" open="{"> <mml:mtable> <mml:mtr> <mml:mtd /> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">ℒ</mml:mi> <mml:mrow> <mml:mi mathvariant="italic">ε</mml:mi> </mml:mrow> </mml:msub> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="normal">=</mml:mo> <mml:mi mathvariant="italic">f</mml:mi> <mml:mo mathvariant="normal">(</mml:mo> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="normal">)</mml:mo> <mml:mo /> <mml:mo /> <mml:mrow> <mml:mi mathvariant="normal">in</mml:mi> </mml:mrow> <mml:mo /> <mml:mo /> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">IR</mml:mi> </mml:mrow> <mml:mrow> <mml:mn mathvariant="normal">3</mml:mn> </mml:mrow> </mml:msup> <mml:mi mathvariant="italic">, </mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="italic">></mml:mo> <mml:mn mathvariant="normal">0</mml:mn> <mml:mo /> <mml:mo /> <mml:mrow> <mml:mi mathvariant="normal">in</mml:mi> </mml:mrow> <mml:mo /> <mml:mo /> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">IR</mml:mi> </mml:mrow> <mml:mrow> <mml:mn mathvariant="normal">3</mml:mn> </mml:mrow> </mml:msup> <mml:mi mathvariant="italic">, </mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="normal">∈</mml:mo> <mml:msup> <mml:mi mathvariant="italic">H</mml:mi> <mml:mrow> <mml:mn mathvariant="normal">1</mml:mn> </mml:mrow> </mml:msup> <mml:mo mathvariant="normal">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">IR</mml:mi> </mml:mrow> <mml:mrow> <mml:mn mathvariant="normal">3</mml:mn> </mml:mrow> </mml:msup> <mml:mo mathvariant="normal">)</mml:mo> <mml:mi mathvariant="italic">, </mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mfenced> </mml:mrow> </mml:math> </alternatives> </disp-formula> </p> <p>where <italic>ε</italic> is a small positive parameter, <italic>f</italic> : ℝ → ℝ is a continuous function, <inline-formula><alternatives><tex-math id="tex_eq3"><![CDATA[$$ \mathcal{L}_{\var} $$]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjx9jvr" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq3" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">ℒ</mml:mi><mml:mi mathvariant="italic">ε</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> is a nonlocal operator defined by</p> <p> <disp-formula> <alternatives> <tex-math id="tex_eq2"><![CDATA[$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$]]></tex-math> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2zjx9jw9" xmlns:xlink="http://www.w3.org/1999/xlink" /> <mml:math id="mml_eq2" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">ℒ</mml:mi> <mml:mi mathvariant="italic">ε</mml:mi> </mml:msub> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="normal">=</mml:mo> <mml:mi mathvariant="italic">M</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mrow> <mml:mfrac> <mml:mn mathvariant="normal">1</mml:mn> <mml:mi mathvariant="italic">ε</mml:mi> </mml:mfrac> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal">∫</mml:mi> <mml:msup> <mml:mi mathvariant="normal">IR</mml:mi> <mml:mn mathvariant="normal">3</mml:mn> </mml:msup> </mml:msub> <mml:mo mathvariant="normal">|</mml:mo> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi mathvariant="italic">u</mml:mi> <mml:msup> <mml:mo mathvariant="normal">|</mml:mo> <mml:mn mathvariant="normal">2</mml:mn> </mml:msup> <mml:mo mathvariant="normal">+</mml:mo> <mml:mrow> <mml:mfrac> <mml:mn mathvariant="normal">1</mml:mn> <mml:msup> <mml:mi mathvariant="italic">ε</mml:mi> <mml:mn mathvariant="normal">3</mml:mn> </mml:msup> </mml:mfrac> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal">∫</mml:mi> <mml:msup> <mml:mi mathvariant="normal">IR</mml:mi> <mml:mn mathvariant="normal">3</mml:mn> </mml:msup> </mml:msub> <mml:mi mathvariant="italic">V</mml:mi> <mml:mo mathvariant="normal">(</mml:mo> <mml:mi mathvariant="italic">x</mml:mi> <mml:mo mathvariant="normal">)</mml:mo> <mml:msup> <mml:mi mathvariant="italic">u</mml:mi> <mml:mn mathvariant="normal">2</mml:mn> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mo mathvariant="normal">[</mml:mo> <mml:mo mathvariant="normal">−</mml:mo> <mml:msup> <mml:mi mathvariant="italic">ε</mml:mi> <mml:mn mathvariant="normal">2</mml:mn> </mml:msup> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="normal">+</mml:mo> <mml:mi mathvariant="italic">V</mml:mi> <mml:mo mathvariant="normal">(</mml:mo> <mml:mi mathvariant="italic">x</mml:mi> <mml:mo mathvariant="normal">)</mml:mo> <mml:msup> <mml:mi mathvariant="italic">u</mml:mi> <mml:mo mathvariant="normal">]</mml:mo> </mml:msup> </mml:mrow> <mml:mi mathvariant="italic">, </mml:mi> </mml:mrow> </mml:math> </alternatives> </disp-formula> </p> <p> <italic>M</italic> : IR<sub>+</sub> → IR<sub>+</sub> and <italic>V</italic> : IR<sup>3</sup> → IR are continuous functions which verify some hypotheses.</p> </abstract>