<p>In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation: <Equation ID="Equ43"> <EquationSource Format="TEX">\(\begin{aligned} -\epsilon ^2\Delta u +\lambda u=\epsilon ^{-(N-\mu )}\left( \int _{\mathbb {R}^N}\frac{Q(y)|u(y)|^p}{|x-y|^{\mu }}{\textrm{d}}y\right) Q(x)|u|^{p-2}u, \text {in}\ \mathbb {R}^N, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <msup> <mi>ϵ</mi> <mn>2</mn> </msup> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>u</mi> <mo>=</mo> <msup> <mi>ϵ</mi> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mfenced close=")" open="("> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </msub> <mfrac> <msup> <mrow> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> </msup> </mfrac> <mtext>d</mtext> <mi>y</mi> </mfenced> <mi>Q</mi> <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> <mtext>in</mtext> <mspace width="4pt" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <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="IEq2"> <EquationSource Format="TEX">\(\mu \in (0,N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\in \left( \frac{2N-\mu }{N}, \frac{2N-\mu }{N-2}\right) \setminus \left\{ \frac{2N-\mu +2}{N}\right\} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> <mo>-</mo> <mi>μ</mi> </mrow> <mi>N</mi> </mfrac> <mo>,</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> <mo>-</mo> <mi>μ</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mfenced> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mfenced close="}" open="{"> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> <mo>-</mo> <mi>μ</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>N</mi> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\epsilon &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a small parameter and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\lambda \in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> appears as a Lagrange multiplier. By developing a new variational approach, we show that this problem has a family of normalized multi-bump solutions focused on the isolated part of the local maximum of the potential <i>Q</i>(<i>x</i>) for sufficiently small <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\epsilon &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The asymptotic behavior of the solutions as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\epsilon \rightarrow 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is also explored. It is worth noting that our results encompass the sublinear case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(p&lt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&lt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, which complements some of the previous works.</p>

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Normalized multi-bump solutions for Choquard equation involving sublinear case

  • He Zhang,
  • Shuai Yao,
  • Haibo Chen

摘要

In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation: \(\begin{aligned} -\epsilon ^2\Delta u +\lambda u=\epsilon ^{-(N-\mu )}\left( \int _{\mathbb {R}^N}\frac{Q(y)|u(y)|^p}{|x-y|^{\mu }}{\textrm{d}}y\right) Q(x)|u|^{p-2}u, \text {in}\ \mathbb {R}^N, \end{aligned}\) - ϵ 2 Δ u + λ u = ϵ - ( N - μ ) R N Q ( y ) | u ( y ) | p | x - y | μ d y Q ( x ) | u | p - 2 u , in R N , where \(N\ge 3\) N 3 , \(\mu \in (0,N)\) μ ( 0 , N ) , \(p\in \left( \frac{2N-\mu }{N}, \frac{2N-\mu }{N-2}\right) \setminus \left\{ \frac{2N-\mu +2}{N}\right\} \) p 2 N - μ N , 2 N - μ N - 2 \ 2 N - μ + 2 N , \(\epsilon >0\) ϵ > 0 is a small parameter and \(\lambda \in \mathbb {R}\) λ R appears as a Lagrange multiplier. By developing a new variational approach, we show that this problem has a family of normalized multi-bump solutions focused on the isolated part of the local maximum of the potential Q(x) for sufficiently small \(\epsilon >0\) ϵ > 0 . The asymptotic behavior of the solutions as \(\epsilon \rightarrow 0\) ϵ 0 is also explored. It is worth noting that our results encompass the sublinear case \(p<2\) p < 2 , which complements some of the previous works.