On the solvability of a boundary value problem on the real line

• Main Authors: Marcelli Cristina, Papalini Francesca, Cupini Giovanni
• Format: Article
• Language: English
• Published: SpringerOpen 2011-01-01
• Series: Boundary Value Problems
• Subjects: boundary value problems; unbounded domains; heteroclinic solutions; nonlinear differential operators; <it>p</it>-Laplacian operator; &#934;-Laplacian operator
• Online Access: http://www.boundaryvalueproblems.com/content/2011/1/26
• ISSN: 1687-2762; 1687-2770

<p>Abstract</p> <p>We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations</p> <p><display-formula><m:math name="1687-2770-2011-26-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo>&#934;</m:mo> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>&#8242;</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>&#8242;</m:mi> </m:mrow> </m:msup> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>x</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>&#8242;</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:mspace width="1em" class="quad"/> <m:mstyle mathvariant="normal"> <m:mi>a</m:mi> </m:mstyle> <m:mstyle mathvariant="normal"> <m:mo class="MathClass-punc">.</m:mo> <m:mi>e</m:mi> </m:mstyle> <m:mstyle mathvariant="normal"> <m:mo class="MathClass-punc">.</m:mo> </m:mstyle> <m:mspace width="2.77695pt" class="tmspace"/> <m:mi>t</m:mi> <m:mo class="MathClass-rel">&#8712;</m:mo> <m:mi>&#8477;</m:mi> </m:mrow> </m:math></display-formula></p> <p>governed by a nonlinear differential operator &#934; extending the classical <it>p-</it>Laplacian, with right-hand side <it>f </it>having the critical rate of decay -1 as |<it>t</it>| &#8594; +&#8734;, that is <inline-formula><m:math name="1687-2770-2011-26-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mo class="MathClass-bin">&#8901;</m:mo> <m:mo class="MathClass-punc">,</m:mo> <m:mo class="MathClass-bin">&#8901;</m:mo> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">&#8776;</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:mfrac> </m:math></inline-formula>. We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure <it>f</it>(<it>t</it>, <it>x</it>, <it>x'</it>) = <it>b</it>(<it>t</it>, <it>x</it>)<it>c</it>(<it>x</it>, <it>x'</it>).</p> <p><b>Mathematical subject classification: Primary: </b>34B40; 34C37; <b>Secondary: </b>34B15; 34L30.</p>