<p>In this paper, we study the following nonlinear magnetic Schrödinger equation<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left( \frac{\varepsilon }{i}\nabla -A(x)\right) ^2u+V(x)u=\lambda f(|u|)u+|u|^{2^*-2}u,\quad x\in \mathbb {R}^N,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mfenced close=")" open="("> <mfrac> <mi>ε</mi> <mi>i</mi> </mfrac> <mi mathvariant="normal">∇</mi> <mo>-</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mn>2</mn> </msup> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a small parameter, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$V(x):\mathbb {R}^{N}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A(x):\mathbb {R}^{N}\rightarrow \mathbb {R}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> are electric and magnetic potentials, respectively. Under a global assumption on the potential <i>V</i>, by using variational methods and Ljusternik–Schnirelmann theory, we prove the existence and multiplicity of solutions for sufficiently large <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and small <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>.</p>

Existence and multiplicity results for the magnetic Schrödinger equation with critical growth

期刊论文
发表日期：

2026年7月31日

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Houzhi Tang,
Ziqing Zhu

摘要

In this paper, we study the following nonlinear magnetic Schrödinger equation $\left( \frac{\varepsilon }{i}\nabla -A(x)\right) ^2u+V(x)u=\lambda f(|u|)u+|u|^{2^*-2}u,\quad x\in \mathbb {R}^N,$ ${(\frac{ε}{i} \nabla - A (x))}^{2} u + V (x) u = λ f (| u |) u + {| u |}^{2^{*} - 2} u, x \in R^{N},$ where $\varepsilon >0$ $ε > 0$ is a small parameter, $\lambda >0$ $λ > 0$ , $N\ge 3$ $N \geq 3$ , $V(x):\mathbb {R}^{N}\rightarrow \mathbb {R}$ $V (x) : R^{N} \to R$ and $A(x):\mathbb {R}^{N}\rightarrow \mathbb {R}^{N}$ $A (x) : R^{N} \to R^{N}$ are electric and magnetic potentials, respectively. Under a global assumption on the potential V, by using variational methods and Ljusternik–Schnirelmann theory, we prove the existence and multiplicity of solutions for sufficiently large $\lambda $ $λ$ and small $\varepsilon $ $ε$ .