<p>In this paper, we consider the semilinear elliptic equation <Equation ID="Equ89"> <EquationSource Format="TEX">$\begin{aligned} \varepsilon ^2\Delta u+V(x)(|u-1|^p-1)=0,~u>0,~u\in H^1(\mathbb {R}^2), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>ε</mi> <mn>2</mn> </msup> <msup> <mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>u</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>u</mi> <mo>∈</mo> </mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <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, the power <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <i>V</i> is a smooth positive function. Under the appropriate gap condition, the problem admits a solution <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> concentrating along a closed curve <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>, which is stationary and nondegenerate with respect to the weighted length functional <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\int _\Gamma V^{\frac{1}{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∫</mo> <mi mathvariant="normal">Γ</mi> </msub> <msup> <mi>V</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

Solutions concentrating on curves for the elliptic equation in $\mathbb {R}^2$

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发表日期：

2026年5月6日

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Weihong Xie,
Mingzhu Yu,
Jiaojiao Zhang

摘要

In this paper, we consider the semilinear elliptic equation $\begin{aligned} \varepsilon ^2\Delta u+V(x)(|u-1|^p-1)=0,~u>0,~u\in H^1(\mathbb {R}^2), \end{aligned}$ $\begin{matrix} ε^{2} {Δ u + V (x) (| u - 1 |}^{p} - 1) = 0, u > 0, u \in H^{1} (R^{2}), \end{matrix}$ where $\varepsilon >0$ $ε > 0$ is a small parameter, the power $p>1$ $p > 1$ , V is a smooth positive function. Under the appropriate gap condition, the problem admits a solution $u_\varepsilon $ $u_{ε}$ concentrating along a closed curve $\Gamma $ $Γ$ , which is stationary and nondegenerate with respect to the weighted length functional $\int _\Gamma V^{\frac{1}{2}}$ $\int_{Γ} V^{\frac{1}{2}}$ .

Solutions concentrating on curves for the elliptic equation in \(\mathbb {R}^2\)

摘要