Fractional <i>p</i> ( <i>x</i> )-Laplacian eigenvalues problems involving <i>q</i> ( <i>x</i> )-Hardy potential

Abstract The aim of this work is to investigate the spectrum of a singular eigenvalue boundary problems involving the fractional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> p(x) -Laplacian operator. We first establish a new <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>;</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(p(x);q(x))} -Hardy inequality related to this operator. Then, using the theory of the fractional Sobolev space with variable exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {W^{s,p(x,y)}(\Omega)} and the variational technique based on the Ljusternik–Schnirelmann theory on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {C^{1}} -manifolds, we prove the existence of at least one unbounded, nondecreasing sequence of nonnegative eigenvalues <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>λ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {(\lambda_{k})_{k\geq 1}} . Finally, we provide sufficient conditions that guarantee the positivity of the infimum of the set Λ of eigenvalues.