This paper focuses on the following homogeneous elliptic system with critical growth<Equation ID="Equ1"> <EquationSource Format="TEX">$\begin{cases}-\Delta_p u + V(x)|u|^{p-2}u = \frac{1}{p^} Q_u(u,v) & \text{in} \mathbb{R}^N, \\-\Delta_p v + W(x)|v|^{p-2}v = \frac{1}{p^} Q_v(u,v) & \text{in} \mathbb{R}^N,\end{cases}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>{</mo> <mtable> <mtr> <mtd> <mo>−</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mo>∗</mo> </msup> </mfrac> <msub> <mi>Q</mi> <mi>u</mi> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mtext>in </mtext> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mi>v</mi> <mo>+</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>v</mi> <mo>=</mo> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mo>∗</mo> </msup> </mfrac> <msub> <mi>Q</mi> <mi>v</mi> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mtext>in </mtext> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" /> </mrow> </math></EquationSource> </Equation>where 1 < p < N, p* = pN/(N − p), and V, W: ℝN → ℝ are two sign-changing functions. By applying a variant of the second concentration-compactness principle, we demonstrate the existence of a mountain-pass solution for the above system. Moreover, we combine a recent global compactness result by Cintra and Correia [7] with Krasnoselskii’s genus theory to demonstrate that the system has at least N distinct pairs of non-trivial solutions in the case of small perturbations of the potentials.

Existence and multiplicity of solutions for homogeneous elliptic systems with critical growth

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发表日期：

2026年7月19日

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Jeziel N. Correia,
Shengda Zeng

摘要

This paper focuses on the following homogeneous elliptic system with critical growth

$\begin{cases}-\Delta_p u + V(x)|u|^{p-2}u = \frac{1}{p^*} Q_u(u,v) & \text{in} \mathbb{R}^N, \\-\Delta_p v + W(x)|v|^{p-2}v = \frac{1}{p^*} Q_v(u,v) & \text{in} \mathbb{R}^N,\end{cases}$ ${\begin{matrix} - Δ_{p} u + V (x) | u |^{p - 2} u = \frac{1}{p^{*}} Q_{u} (u, v) & in R^{N}, \\ - Δ_{p} v + W (x) | v |^{p - 2} v = \frac{1}{p^{*}} Q_{v} (u, v) & in R^{N}, \end{matrix}$

where 1 < p < N, p* = pN/(N − p), and V, W: ℝ^N → ℝ are two sign-changing functions. By applying a variant of the second concentration-compactness principle, we demonstrate the existence of a mountain-pass solution for the above system. Moreover, we combine a recent global compactness result by Cintra and Correia [7] with Krasnoselskii’s genus theory to demonstrate that the system has at least N distinct pairs of non-trivial solutions in the case of small perturbations of the potentials.