<abstract xmlns="http://www.w3.org/1999/xhtml">

In the present paper, we prove the existence of the solutions (<italic>λ, u</italic>) ∈ ℝ <italic>× H</italic>1(ℝ3) to the following Kirchhoff equations with the Hartree-type nonlinearity under the general mass supercritical settings,
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2023-0015_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>-</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>ℝ</mi></mrow><mn>3</mn></msup></mrow></munder><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mo>∇</mo><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mn>2</mn></msup><mi>d</mi><mi>x</mi></mrow></mrow></mrow><mo>)</mo></mrow><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>-</mo><mi>λ</mi><mi>u</mi><mo>=</mo><mrow><mo>[</mo> <mrow><msub><mrow><mi>I</mi></mrow><mi>α</mi></msub><mo>*</mo><mrow><mo>(</mo><mrow><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mn>1</mn></msup><mrow><mo>(</mo><mrow><msup><mrow><mi>ℝ</mi></mrow><mn>3</mn></msup></mrow><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math>
<tex-math>\left\{ {\matrix{{ - \left( {a + b\int\limits_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u - \lambda u = \left[ {{I_\alpha }*\left( {K\left( x \right)F\left( u \right)} \right)} \right]K\left( x \right)f\left( u \right),} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^3}} \right),} \hfill \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
where <italic>a, b ></italic> 0 are prescribed, <italic>I</italic><italic>α</italic> = <italic>|x|</italic><italic>α</italic>−3 is the riesz potential where <italic>α</italic> ∈ (0, 3), <italic>K</italic> ∈ 𝒞1(ℝ3, ℝ+) satisfies an explicit assumption and <italic>f</italic> ∈𝒞 (ℝ, ℝ) satisfies some weak conditions, we develop some new tricks for dealing with the Hartree-type term to overcome the difficulties produced by the appearance of non-constant potential <italic>K</italic>(<italic>x</italic>). This paper extends and promotes the previous results on prescribed <italic>L</italic>2-norm solutions of the Kirchhoff-type equation.
</abstract>

In the present paper, we prove the existence of the solutions (λ, u) ∈ ℝ × H1(ℝ3) to the following Kirchhoff equations with the Hartree-type nonlinearity under the general mass supercritical settings,



{ -(a+b∫ℝ3| ∇u |2dx)Δu-λu=[ Iα*(K(x)F(u)) ]K(x)f(u),u∈H1(ℝ3),
\left\{ {\matrix{{ - \left( {a + b\int\limits_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u - \lambda u = \left[ {{I_\alpha }*\left( {K\left( x \right)F\left( u \right)} \right)} \right]K\left( x \right)f\left( u \right),} \hfill  \cr {u \in {H^1}\left( {{\mathbb{R}^3}} \right),} \hfill  \cr  } } \right.


where a, b > 0 are prescribed, Iα = |x|α−3 is the riesz potential where α ∈ (0, 3), K ∈ 𝒞1(ℝ3, ℝ+) satisfies an explicit assumption and f ∈𝒞 (ℝ, ℝ) satisfies some weak conditions, we develop some new tricks for dealing with the Hartree-type term to overcome the difficulties produced by the appearance of non-constant potential K(x). This paper extends and promotes the previous results on prescribed L2-norm solutions of the Kirchhoff-type equation.

Normalized solutions of Kirchhoff equations with Hartree-type nonlinearity

Hebei Vocational University of Industry and Technology

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Normalized solutions of Kirchhoff equations with Hartree-type nonlinearity"}

In the present paper, we prove the existence of the solutions (λ, u) ∈ ℝ × H1(ℝ3) to the following Kirchhoff equations with the Hartree-type nonlinearity under...

Normalized solutions of Kirchhoff equations with Hartree-type nonlinearity

Published Online: Feb 04, 2023

Page range: 271 - 294

Received: Apr 13, 2022

Accepted: Jun 15, 2022

DOI: https://doi.org/10.2478/auom-2023-0015

Keywords
Kirchhoff problem, Normalized solution, Hartree-type nonlinearity, mass supercritical case

© 2023 Shuai Yuan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Normalized solutions of Kirchhoff equations with Hartree-type nonlinearity

Published Online: Feb 04, 2023

Page range: 271 - 294

Received: Apr 13, 2022

Accepted: Jun 15, 2022

DOI: https://doi.org/10.2478/auom-2023-0015

KeywordsKirchhoff problem, Normalized solution, Hartree-type nonlinearity, mass supercritical case

© 2023 Shuai Yuan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Kirchhoff problem, Normalized solution, Hartree-type nonlinearity, mass supercritical case