A topological analysis of <i>p</i> ( <i>x</i> )-harmonic functionals in one-dimensional nonlocal elliptic equations

Abstract We consider a class of one-dimensional elliptic equations possessing a p ( x )-harmonic functional as a nonlocal coefficient. As part of our results we treat the model case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mo>−</m:mo> <m:mi>M</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:munderover accent="true" accentunder="false"> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mspace width="0.3333em"/> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mi>f</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mtext>, </m:mtext> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>t</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:math> $${-}M\left(\underset{0{}}{\overset{1}{\int }}{\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}\ \mathrm{d}x\right){u}^{\prime \prime }(t)=\lambda f\left(t,u(t)\right)\text{,\hspace{0.17em}}0&lt; t&lt; 1$$ subject to the boundary data <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>=</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> $$u(0)=0=u(1).$$ In addition, we consider a broader class of problems, of which the model case in a special case, by writing the argument of M as a finite convolution. As part of the analysis, a simple but fundamental lemma in introduced that allows the estimation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> ${\left\vert {u}^{\prime }(x)\right\vert }^{p(x)}$ in terms of constant exponents; this is the key to circumventing the variable exponent. An unusual array of analytical tools is used, including Sobolev’s inequality. Our results address both existence and nonexistence of solution.