<p>We obtain multiple solutions for the zero mass Schrödinger–Poisson–Slater equation <Equation ID="Equ27"> <EquationSource Format="TEX">$ - \Delta u + \left( \frac{1}{4 \pi | x |} *u^2 \right) u = \lambda g (x) | u |^{p - 2} u + | u |^{6 - 2} u \text {, }\qquad u \in \mathcal {D}^{1, 2} (\mathbb {R}^3) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> <mrow /> <mo>∗</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mfenced> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mi>λ</mi> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>6</mn> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mtext>,</mtext> <mspace width="0.333333em" /> <mspace width="2em" /> <mi>u</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda \gg 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≫</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \in (4, 6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g \in L^{6 / (6 - p)} (\mathbb {R}^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow> <mn>6</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mo>-</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The crucial <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\operatorname {PS})_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mo>PS</mo> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation> condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.</p>

Multiple solutions for Schrödinger–Poisson–Slater equations with critical growth

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发表日期：

2026年4月8日

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Shibo Liu

摘要

We obtain multiple solutions for the zero mass Schrödinger–Poisson–Slater equation $ - \Delta u + \left( \frac{1}{4 \pi | x |} *u^2 \right) u = \lambda g (x) | u |^{p - 2} u + | u |^{6 - 2} u \text {, }\qquad u \in \mathcal {D}^{1, 2} (\mathbb {R}^3) $ $- Δ u + (\frac{1}{4 π | x |} * u^{2}) u = {λ g (x) | u |}^{p - 2} u + {| u |}^{6 - 2} u, u \in D^{1, 2} (R^{3})$ for $\lambda \gg 1$ $λ ≫ 1$ , where $p \in (4, 6)$ $p \in (4, 6)$ and $g \in L^{6 / (6 - p)} (\mathbb {R}^3)$ $g \in L^{6 / (6 - p)} (R^{3})$ . The crucial $(\operatorname {PS})_c$ ${(PS)}_{c}$ condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.