<p>We consider the equation <Equation ID="Equ19"> <EquationSource Format="TEX">$\begin{aligned} -\Delta u+u=Q_\varepsilon (x)|u|^{p-2}u,\quad u\in H^1(\mathbb {R}^N), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mo>=</mo> <msub> <mi>Q</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <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> <mspace width="1em" /> <mi>u</mi> <mo>∈</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q_\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> takes the value 1 on each ball <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_\varepsilon (y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$y\in \mathbb {Z}^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and the value <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> elsewhere. We establish the existence of a least energy solution for each <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon \in (0,\frac{1}{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and show that their <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> norms concentrate locally at points of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {Z}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core

期刊论文
出版模式：

开放获取
发表日期：

2025年11月21日

查看全文

Mónica Clapp,
Alberto Saldaña,
Andrzej Szulkin

摘要

We consider the equation $\begin{aligned} -\Delta u+u=Q_\varepsilon (x)|u|^{p-2}u,\quad u\in H^1(\mathbb {R}^N), \end{aligned}$ $\begin{matrix} - Δ u + u = Q_{ε} (x) {| u |}^{p - 2} u, u \in H^{1} (R^{N}), \end{matrix}$ where $Q_\varepsilon $ $Q_{ε}$ takes the value 1 on each ball $B_\varepsilon (y)$ $B_{ε} (y)$ , $y\in \mathbb {Z}^N$ $y \in Z^{N}$ , and the value $-1$ $- 1$ elsewhere. We establish the existence of a least energy solution for each $\varepsilon \in (0,\frac{1}{2})$ $ε \in (0, \frac{1}{2})$ and show that their $H^1$ $H^{1}$ and $L^p$ $L^{p}$ norms concentrate locally at points of $\mathbb {Z}^N$ $Z^{N}$ as $\varepsilon \rightarrow 0$ $ε \to 0$ .