Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros. Issue 10 (17th January 2014)

• Record Type: Journal Article
• Authors: Iturriaga, Leonelo; Massa, Eugenio; Sánchez, Justino; Ubilla, Pedro
• Is Part Of: Mathematische Nachrichten. Volume 287:Issue 10(2014)
• Page Start: 1131
• Page End: 1141
• Publication Date: 2014-01-17
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201100285
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We study existence, multiplicity, and the behavior, with respect to λ, of positive radially symmetric solutions of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8sbcsd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201100285:equation:mana201100285-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>h</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> in annular domains in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8sbcrv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201100285:equation:mana201100285-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>N</mml:mi></mml:msup><mml:mo>, </mml:mo><mml:mspace width="0.28em" /><mml:mspace width="0.28em" /><mml:mi>N</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. The nonlinear term has a superlinear local growth at infinity, is nonnegative, and satisfies <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8sbcq9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201100285:equation:mana201100285-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>h</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>a</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width="0.16em" /></mml:mrow></mml:math></alternatives></inline-formula> for a suitable positive and concave function <italic>a</italic>. For this, we combine several methods such as the sub and supersolutions method, a priori estimates and degree theory.</p> </abstract>