<p>In this paper, we investigate the existence of nontrivial solutions to the following critical nonhomogeneous Choquard equation: <Equation ID="Equa"><EquationSource Format="MATHML"><math><mrow><mo>{</mo><mtable columnalign="right left" columnspacing="0.2em"><mtr><mtd><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>u</mi></mtd><mtd><mo>=</mo><mi>λ</mi><mi>u</mi><mo>+</mo><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mo>(</mo><mspace width="0.25em" /><mfrac><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">|</mo><msubsup><mn>2</mn><mi>α</mi><mo>∗</mo></msubsup></msup></mrow><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">|</mo><mi>α</mi></msup></mrow></mfrac><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mspace width="0.25em" /><mo stretchy="false">|</mo><mi>u</mi><msup><mo stretchy="false">|</mo><mrow><msubsup><mn>2</mn><mi>α</mi><mo>∗</mo></msubsup><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em" /><mi>i</mi><mi>n</mi><mspace width="1em" /><mi mathvariant="normal">Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>∈</mo></mtd><mtd><mspace width="0.2em" /><msubsup><mi>H</mi><mrow><mn>0</mn></mrow><mn>1</mn></msubsup><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></math></EquationSource><EquationSource Format="TEX">$ \left \{ \begin{aligned} -\Delta u&=\lambda u +\int _{\Omega }\left (\ \frac{|u(y)|^{2^{}_{\alpha }}}{|x-y|^{\alpha }}dy\right )\ |u|^{2^{}_{\alpha }-2}u + f(x) \quad in \quad \Omega ,\\ u \in &\, H^{1}_{0}(\Omega ), \end{aligned}\right . $</EquationSource></Equation> where <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi>N</mi><mo>≥</mo><mn>4</mn></math></EquationSource><EquationSource Format="TEX">$N\geq 4$</EquationSource></InlineEquation>, <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><mi>λ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></math></EquationSource><EquationSource Format="TEX">$\lambda \in \mathbb{R}$</EquationSource></InlineEquation>, <InlineEquation ID="IEq3"><EquationSource Format="MATHML"><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>N</mi></math></EquationSource><EquationSource Format="TEX">$0<\alpha <N$</EquationSource></InlineEquation> and <InlineEquation ID="IEq4"><EquationSource Format="MATHML"><math><msubsup><mn>2</mn><mi>α</mi><mo>∗</mo></msubsup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi><mo>−</mo><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></EquationSource><EquationSource Format="TEX">$2^{*}_{\alpha }=\frac{2N-\alpha }{N-2}$</EquationSource></InlineEquation> is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By employing an abstract critical point theorem, we establish the existence of two distinct nontrivial solutions to the above problem when <InlineEquation ID="IEq5"><EquationSource Format="MATHML"><math><mi>λ</mi><mo>≥</mo><msub><mi>λ</mi><mn>1</mn></msub></math></EquationSource><EquationSource Format="TEX">$\lambda \geq \lambda _{1}$</EquationSource></InlineEquation>. Our results extend results in the literature for <InlineEquation ID="IEq6"><EquationSource Format="MATHML"><math><mn>0</mn><mo><</mo><mi>λ</mi><mo><</mo><msub><mi>λ</mi><mn>1</mn></msub></math></EquationSource><EquationSource Format="TEX">$0<\lambda <\lambda _{1}$</EquationSource></InlineEquation>.</p>

On nonhomogeneous Choquard equation with Hardy–Littlewood–Sobolev critical nonlinearity

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2026年6月15日

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Rachid Echarghaoui,
Rachid Kouik,
Rachid Sersif

摘要

In this paper, we investigate the existence of nontrivial solutions to the following critical nonhomogeneous Choquard equation: ${\begin{aligned} - Δ u & = λ u + \int_{Ω} (\frac{| u (y) |^{2_{α}^{*}}}{| x - y |^{α}} d y) | u |^{2_{α}^{*} - 2} u + f (x) i n Ω, \\ u \in & H_{0}^{1} (Ω), \end{aligned}$ $ \left \{ \begin{aligned} -\Delta u&=\lambda u +\int _{\Omega }\left (\ \frac{|u(y)|^{2^{*}_{\alpha }}}{|x-y|^{\alpha }}dy\right )\ |u|^{2^{*}_{\alpha }-2}u + f(x) \quad in \quad \Omega ,\\ u \in &\, H^{1}_{0}(\Omega ), \end{aligned}\right . $ where $N \geq 4$ $N\geq 4$, $λ \in R$ $\lambda \in \mathbb{R}$, $0 < α < N$ $0<\alpha <N$ and $2_{α}^{*} = \frac{2 N - α}{N - 2}$ $2^{*}_{\alpha }=\frac{2N-\alpha }{N-2}$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By employing an abstract critical point theorem, we establish the existence of two distinct nontrivial solutions to the above problem when $λ \geq λ_{1}$ $\lambda \geq \lambda _{1}$. Our results extend results in the literature for $0 < λ < λ_{1}$ $0<\lambda <\lambda _{1}$.