<p>In this work, we study a class of Kirchhoff-type equations with indefinite nonlinearities of the form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$-\left( 1+\lambda \int _{\mathbb {R}^3} |\nabla u|^2\right) \Delta u = f(x)|u|^{\gamma -2}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mfenced close=")" open="("> <mn>1</mn> <mo>+</mo> <mi>λ</mi> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mfenced> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>γ</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a parameter. Using variational methods and a simplified version of the concentration-compactness principle, we overcome the lack of compactness inherent to the unbounded domain. Under suitable conditions on <i>f</i>, we provide a precise description of the bifurcation phenomena occurring with respect to the parameter <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. We show that there exists an extremal parameter <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda ^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>λ</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> which acts as a turning point where two branches of solutions collapse and vanish. Our results generalize previous studies by removing the boundedness condition on the domain and considering sign-changing nonlinearities.</p>

Existence and multiplicity of solutions for a Kirchhoff-type problem with indefinite nonlinearity in $\mathbb {R}^3$

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2026年5月21日

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Steffânio Moreno de Sousa

摘要

In this work, we study a class of Kirchhoff-type equations with indefinite nonlinearities of the form $-\left( 1+\lambda \int _{\mathbb {R}^3} |\nabla u|^2\right) \Delta u = f(x)|u|^{\gamma -2}u$ $- (1 + λ \int_{R^{3}} {| \nabla u |}^{2}) Δ u = f (x) {| u |}^{γ - 2} u$ in $\mathbb {R}^3$ $R^{3}$ , where $\lambda >0$ $λ > 0$ is a parameter. Using variational methods and a simplified version of the concentration-compactness principle, we overcome the lack of compactness inherent to the unbounded domain. Under suitable conditions on f, we provide a precise description of the bifurcation phenomena occurring with respect to the parameter $\lambda $ $λ$ . We show that there exists an extremal parameter $\lambda ^*$ $λ^{*}$ which acts as a turning point where two branches of solutions collapse and vanish. Our results generalize previous studies by removing the boundedness condition on the domain and considering sign-changing nonlinearities.

Existence and multiplicity of solutions for a Kirchhoff-type problem with indefinite nonlinearity in \(\mathbb {R}^3\)

摘要