Multiplicity of solutions for singular superlinear Kirchhoff equations involving indefinite potentials

We are concerned with the multiplicity of solutions for the singular superlinear Kirchhoff equation involving indefinite potentials <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mo>−</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:mo stretchy="false">|</m:mo> <m:mi>∇</m:mi> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mtext>d</m:mtext> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</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:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mi>g</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:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>γ</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>h</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:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mtext>in</m:mtext> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:mspace width="0.17em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> $$-(1+b{{R}³}{∫ } ∇ u{ }²dx){Δ}u+V(x)u=λ g(x){u}-γ+h(x){u}ᵖ in {R}³,$$ when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> 0< g∈ L^γ ₀(R³) , the potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> h∈ L∞(R³) may change its sign, V is a positive continuous function, and λ > 0 is a real parameter. The main difficulties come from the non-differentiability of the energy functional and the fact that the Nehari extremal value is not available. We overcome these difficulties by analyzing the structure of the Nehari manifold and redefine the range for the existence of solutions, which does not depend on the Nehari extremal value. As a result, two different solutions of that problem are obtained with negative and positive energy, respectively.