New class of sixth-order nonhomogeneous <i>p</i> ( <i>x</i> )-Kirchhoff problems with sign-changing weight functions
Mohamed Karim Hamdani, Nguyen Thanh Chung, Dušan D. Repovš
Abstract
Abstract In this paper, we prove the existence of multiple solutions for the following sixth-order p ( x )-Kirchhoff-type problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfenced open="{" close=""> <m:mtable columnalign="left left" rowspacing="0.4em 0.1em" columnspacing="1em"> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:mi>M</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:munder> <m:mo>∫</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Ω</m:mi> </m:mrow> </m:munder> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi mathvariant="normal">∇</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> <m:mi>d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msubsup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>3</m:mn> </m:msubsup> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>r</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>h</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mtd> <m:mtd> <m:mtext>in</m:mtext> <m:mspace width="1em"/> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Ω</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn>