<p>This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff–Boussinesq-type problems given by <Equation ID="Equ41"> <EquationSource Format="TEX">$ \left\{ \begin{array}{l} \Delta (a(\varepsilon x) \Delta u) - \Delta _p u + u = Q_{u}(u,v) + \dfrac{1}{2_{}} K_{u}(u,v) \;\; \text {in} \;\; \mathbb {R}^N,\\ \Delta (b(\varepsilon x) \Delta v) - \Delta _p v + v = Q_{v}(u,v) + \dfrac{1}{2_{}} K_{v}(u,v) \;\; \text {in} \;\; \mathbb {R}^N,\\ u, v \in H^{2}(\mathbb {R}^N), \end{array} \right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>ε</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mo>=</mo> <msub> <mi>Q</mi> <mi>u</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mn>2</mn> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> </mfrac> </mstyle> <msub> <mi>K</mi> <mi>u</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mtext>in</mtext> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>ε</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>v</mi> <mo>+</mo> <mi>v</mi> <mo>=</mo> <msub> <mi>Q</mi> <mi>v</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mn>2</mn> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> </mfrac> </mstyle> <msub> <mi>K</mi> <mi>v</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mtext>in</mtext> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$2< p < 2^{}= \frac{2N}{N-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <msup> <mn>2</mn> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2_{}= \frac{2N}{N-4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mn>2</mn> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>4</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. We establish the existence of a ground state solution for the critical system by employing variational methods, and we prove the multiplicity of solutions through the Ljusternik–Schnirelmann category.</p>

Existence and multiplicity of solutions for a critical system involving a biharmonic and p-Laplacian operator

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2026年7月16日

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R. D. Carlos,
S. M. A. Salirrosas

摘要

This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff–Boussinesq-type problems given by $ \left\{ \begin{array}{l} \Delta (a(\varepsilon x) \Delta u) - \Delta _p u + u = Q_{u}(u,v) + \dfrac{1}{2_{*}} K_{u}(u,v) \;\; \text {in} \;\; \mathbb {R}^N,\\ \Delta (b(\varepsilon x) \Delta v) - \Delta _p v + v = Q_{v}(u,v) + \dfrac{1}{2_{*}} K_{v}(u,v) \;\; \text {in} \;\; \mathbb {R}^N,\\ u, v \in H^{2}(\mathbb {R}^N), \end{array} \right. $ $\{\begin{matrix} Δ (a (ε x) Δ u) - Δ_{p} u + u = Q_{u} (u, v) + \frac{1}{2_{*}} K_{u} (u, v) in R^{N}, \\ Δ (b (ε x) Δ v) - Δ_{p} v + v = Q_{v} (u, v) + \frac{1}{2_{*}} K_{v} (u, v) in R^{N}, \\ u, v \in H^{2} (R^{N}), \end{matrix})$ where $\varepsilon >0$ $ε > 0$ , $2< p < 2^{*}= \frac{2N}{N-2}$ $2 < p < 2^{*} = \frac{2 N}{N - 2}$ , $2_{*}= \frac{2N}{N-4}$ $2_{*} = \frac{2 N}{N - 4}$ and $N\ge 5$ $N \geq 5$ . We establish the existence of a ground state solution for the critical system by employing variational methods, and we prove the multiplicity of solutions through the Ljusternik–Schnirelmann category.