In this article, we consider the following fractional (p, q)-Laplacian equation with critical exponent <Equation ID="Equ49"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1944_Article_Equ49.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="428" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} (-\Delta _p)^{s_{1}} u+(-\Delta _q)^{s_{2}} u=\lambda g(x)|u|^{r-2} u+h(x)|u|^{p_{s_{1}}^-2} u \; \text {in}\; {\mathbb {R}}^N, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> </msup> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>s</mi> <mn>2</mn> </msub> </msup> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mi>g</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>r</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mi>h</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> <msubsup> <mi>p</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> <mo>∗</mo> </msubsup> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mspace width="0.277778em" /> <mtext>in</mtext> <mspace width="0.277778em" /> <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="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1944_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$0<s_{2}<s_{1}<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1944_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="144" /> </InlineMediaObject> <EquationSource Format="TEX">$1<q \le p<r< p_{s_{1}}^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>r</mi> <mo><</mo> <msubsup> <mi>p</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1944_Article_IEq5.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$p_{s}^*:=\frac{ Np}{N-ps}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>p</mi> <mrow> <mi>s</mi> </mrow> <mo>∗</mo> </msubsup> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <mi mathvariant="italic">Np</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mi>p</mi> <mi>s</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1944_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$s\in (0,1).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Under certain assumptions on g et h,  using an abstract critical point theorem, we obtain a multiple solutions for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1944_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> sufficiently large. A similar problem with subcritical exponents is also considered.

Multiplicity Results for Fractional (p, q)-Laplacian Equations in ${\mathbb {R}}^N$

期刊论文
发表日期：

2025年8月4日

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Rachid Echarghaoui,
Rachid Sersif

摘要

In this article, we consider the following fractional (p, q)-Laplacian equation with critical exponent $\begin{aligned} (-\Delta _p)^{s_{1}} u+(-\Delta _q)^{s_{2}} u=\lambda g(x)|u|^{r-2} u+h(x)|u|^{p_{s_{1}}^*-2} u \; \text {in}\; {\mathbb {R}}^N, \end{aligned}$ $\begin{matrix} {(- Δ_{p})}^{s_{1}} u + {(- Δ_{q})}^{s_{2}} u = λ g (x) {| u |}^{r - 2} u + h (x) {| u |}^{p_{s_{1}}^{*} - 2} u in R^{N}, \end{matrix}$ where $0<s_{2}<s_{1}<1$ $0 < s_{2} < s_{1} < 1$ , $1<q \le p<r< p_{s_{1}}^*$ $1 < q \leq p < r < p_{s_{1}}^{*}$ and $p_{s}^*:=\frac{ Np}{N-ps}$ $p_{s}^{*} : = \frac{Np}{N - p s}$ for any $s\in (0,1).$ $s \in (0, 1) .$ Under certain assumptions on g et h, using an abstract critical point theorem, we obtain a multiple solutions for $\lambda $ $λ$ sufficiently large. A similar problem with subcritical exponents is also considered.

Multiplicity Results for Fractional (p, q)-Laplacian Equations in \({\mathbb {R}}^N\)

摘要