<p>We study a class of nonlocal Kirchhoff-type problems involving the fractional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ p $</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-Laplacian and critical Sobolev growth. The equation includes a Kirchhoff term <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ M(t) = a + t^m $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>t</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ a \ge 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ m > 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and is posed on a bounded domain with Lipschitz boundary. Using variational methods, Krasnoselskii’s genus theory, and a fractional concentration-compactness principle, we prove the existence of infinitely many weak solutions in both the non-degenerate (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ a > 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and degenerate (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ a = 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) cases.</p>

Infinitely many solutions for a class of fractional Kirchhoff problems with critical exponent

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2026年4月29日

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M. R. Marcial,
L. C. Paes-Leme,
E. M. Martins,
W. M. Ferreira

摘要

We study a class of nonlocal Kirchhoff-type problems involving the fractional $ p $ $p$ -Laplacian and critical Sobolev growth. The equation includes a Kirchhoff term $ M(t) = a + t^m $ $M (t) = a + t^{m}$ , with $ a \ge 0 $ $a \geq 0$ and $ m > 0 $ $m > 0$ , and is posed on a bounded domain with Lipschitz boundary. Using variational methods, Krasnoselskii’s genus theory, and a fractional concentration-compactness principle, we prove the existence of infinitely many weak solutions in both the non-degenerate ( $ a > 0 $ $a > 0$ ) and degenerate ( $ a = 0 $ $a = 0$ ) cases.