Existence results for a class of nonlocal problems involving p-Laplacian
<p>Abstract</p> <p>This paper is concerned with the existence of solutions to a class of p-Kirchhoff type equations with Neumann boundary data as follows:</p> <p> <display-formula> <m:math name="1687-2770-2011-32-i1" xmlns:m="http://www.w3.org/19...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2011-01-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://www.boundaryvalueproblems.com/content/2011/1/32 |
| Summary: | <p>Abstract</p> <p>This paper is concerned with the existence of solutions to a class of p-Kirchhoff type equations with Neumann boundary data as follows:</p> <p> <display-formula> <m:math name="1687-2770-2011-32-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <m:mtable equalrows="false" columnlines="none none none none none none none none none none none none none none none none none none none" equalcolumns="false" class="array"> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mo class="MathClass-bin">-</m:mo> <m:msup> <m:mrow> <m:mfenced separators="" open="[" close="]"> <m:mrow> <m:mi>M</m:mi> <m:mfenced separators="" open="(" close=")"> <m:mrow> <m:msup> <m:mrow> <m:msub> <m:mrow> <m:mo class="MathClass-op">∫ </m:mo> </m:mrow> <m:mrow> <m:mo>Ω</m:mo> </m:mrow> </m:msub> <m:mfenced separators="" open="|" close="|"> <m:mrow> <m:mo class="MathClass-op">∇</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mstyle mathvariant="normal"> <m:mi>d</m:mi> </m:mstyle> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <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>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mstyle mathvariant="normal"> <m:mtext>in</m:mtext> </m:mstyle> <m:mspace width="0.3em" class="thinspace"/> <m:mo>Ω</m:mo> <m:mo class="MathClass-punc">;</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mfrac> <m:mrow> <m:mi>∂</m:mi> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:mi>υ</m:mi> </m:mrow> </m:mfrac> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mstyle mathvariant="normal"> <m:mtext>on</m:mtext> </m:mstyle> <m:mspace width="0.3em" class="thinspace"/> <m:mi>∂</m:mi> <m:mo>Ω</m:mo> <m:mstyle mathvariant="normal"> <m:mo class="MathClass-punc">.</m:mo> </m:mstyle> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula> </p> <p>By means of a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem.</p> |
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| ISSN: | 1687-2762 1687-2770 |