Negative Energy Solutions for a New Fractional<math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"><mi>p</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced></math>-Kirchhoff Problem without the (AR) Condition
Negative Energy Solutions for a New Fractional<math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"><mi>p</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced></math>-Kirchhoff Problem without the (AR) Condition
In this paper, we investigate the following Kirchhoff type problem involving the fractional <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"><mi>p</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced></math> -Laplacian operator. <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"><mfenced open="{" close=""><mrow><mtable class="cases"><mtr><mtd columnalign="left"><mfenced open="(" close=")"><mrow><mi>a</mi><mo>−</mo><mi>b</mi><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi><mo>×</mo><mi>Ω</mi></mrow></msub><mfenced open="(" close=")"><mrow><msup><mrow><mfenced open="|" close="|"><mrow><mi>u</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced><mo>−</mo><mi>u</mi><mfenced open="(" close=")"><mrow><mi>y</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>p</mi><mfenced open="(" close=")"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></msup><mo>/</mo><mi>p</mi><mfenced open="(" close=")"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><msup><mrow><mfenced open="|" close="|"><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mi>s</mi><mi>p</mi><mfenced open="(" close=")"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow></mfenced><mi mathvariant="script">L</mi><mi>u</mi><mo>=</mo><mi>λ</mi><msup><mrow><mfenced open="|" close="|"><mrow><mi>u</mi></mrow></mfenced></mrow><mrow><mi>q</mi><mfenced open="(" close=")"><mrow><mi>x</mi></mrow></mfenced><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>f</mi><mfenced …