<p>In this paper, we consider the following fractional Schrödinger-Poisson system with critical exponent <Equation ID="Equ55"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_Equ55.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="378" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+V_{\lambda } (x)u+\phi u=|u|^{2_s^*-2}u+f(u), &amp; ~\textrm{in}~~\mathbb {R}^3, \\ (-\Delta )^t \phi =u^2, &amp; ~\textrm{in}~~\mathbb {R}^3, \end{array}\right. } \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mo>+</mo> <msub> <mi>V</mi> <mi>λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>ϕ</mi> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <msubsup> <mn>2</mn> <mi>s</mi> <mo>∗</mo> </msubsup> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="3.33333pt" /> <mtext>in</mtext> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msup> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="3.33333pt" /> <mtext>in</mtext> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">\(s\in (\frac{3}{4},1), t\in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(2_s^*:=\frac{6}{3-2s}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mn>2</mn> <mi>s</mi> <mo>∗</mo> </msubsup> <mo>:</mo> <mo>=</mo> <mfrac> <mn>6</mn> <mrow> <mn>3</mn> <mo>-</mo> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is the fractional critical exponent, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(V_{\lambda }(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mi>λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> = <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda V(x)+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Under some suitable assumptions on <i>f</i> and <i>V</i>, if <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is large enough, we prove the existence of ground state sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma. Moreover, the least energy of sign-changing solution is strictly more than twice the energy of the ground state solution. At the same time, we also study the asymptotic behavior of ground state sign-changing solutions as <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1297_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Our results improve the recent results in the literature.</p>

错误:搜索内容不能为空,请输入英文关键词
错误:关键词超出字数限制,请精简
高级检索

Ground State Sign-Changing Solution for a Fractional Schrödinger-Poisson System with Critical Exponent and Steep Potential Well

  • Jiao Fu,
  • Jia-Feng Liao

摘要

In this paper, we consider the following fractional Schrödinger-Poisson system with critical exponent \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+V_{\lambda } (x)u+\phi u=|u|^{2_s^*-2}u+f(u), & ~\textrm{in}~~\mathbb {R}^3, \\ (-\Delta )^t \phi =u^2, & ~\textrm{in}~~\mathbb {R}^3, \end{array}\right. } \end{aligned}\) ( - Δ ) s u + V λ ( x ) u + ϕ u = | u | 2 s - 2 u + f ( u ) , in R 3 , ( - Δ ) t ϕ = u 2 , in R 3 , where \(s\in (\frac{3}{4},1), t\in (0,1)\) s ( 3 4 , 1 ) , t ( 0 , 1 ) , \(2_s^*:=\frac{6}{3-2s}\) 2 s : = 6 3 - 2 s is the fractional critical exponent, \(V_{\lambda }(x)\) V λ ( x ) = \(\lambda V(x)+1\) λ V ( x ) + 1 with \(\lambda >0\) λ > 0 . Under some suitable assumptions on f and V, if \(\lambda >0\) λ > 0 is large enough, we prove the existence of ground state sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma. Moreover, the least energy of sign-changing solution is strictly more than twice the energy of the ground state solution. At the same time, we also study the asymptotic behavior of ground state sign-changing solutions as \(\lambda \rightarrow \infty \) λ . Our results improve the recent results in the literature.