<p>In this paper, we consider the following Kirchhoff–Schrödinger system in <InlineEquation ID="IEq3"> <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="IEq4"> <EquationSource Format="TEX">$a_1, a_2, b_1, b_2, \lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> are positive constants and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a nonnegative constant. In the case where the potentials are constant functions, the second author of the current paper proved that this system admits a positive ground state solution when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$3<p\le q<6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> <mo><</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>(subcritical case) and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$3<p<q=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>(critical case) by using Nehari–Pohozaev manifold. In this paper we extend the result to the nonconstant potential case, that is, under certain assumptions on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$V_1(x), V_2(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we prove that this problem has a nontrivial ground state solution. The main ingredients of the proof are Jeanjean’s monotonicity trick and the global compactness lemma. In particular, we will show that when <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is sufficiently large, the global compactness lemma is valid even in the critical case.</p>

Existence of ground state solutions to Kirchhoff–Schrödinger system for subcritical and critical cases with nonconstant potentials in ${\mathbb {R}}^3$

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发表日期：

2026年2月21日

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Hiroshi Matsuzawa,
Tatsuya Ueno

摘要

In this paper, we consider the following Kirchhoff–Schrödinger system in ${\mathbb {R}}^3$ $R^{3}$ : where $a_1, a_2, b_1, b_2, \lambda $ $a_{1}, a_{2}, b_{1}, b_{2}, λ$ are positive constants and $\mu $ $μ$ is a nonnegative constant. In the case where the potentials are constant functions, the second author of the current paper proved that this system admits a positive ground state solution when $3<p\le q<6$ $3 < p \leq q < 6$ (subcritical case) and $3<p<q=6$ $3 < p < q = 6$ (critical case) by using Nehari–Pohozaev manifold. In this paper we extend the result to the nonconstant potential case, that is, under certain assumptions on $V_1(x), V_2(x)$ $V_{1} (x), V_{2} (x)$ and $\lambda ,$ $λ,$ we prove that this problem has a nontrivial ground state solution. The main ingredients of the proof are Jeanjean’s monotonicity trick and the global compactness lemma. In particular, we will show that when $\mu $ $μ$ is sufficiently large, the global compactness lemma is valid even in the critical case.

Existence of ground state solutions to Kirchhoff–Schrödinger system for subcritical and critical cases with nonconstant potentials in \({\mathbb {R}}^3\)

摘要