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

In this paper, we firstly explore the existence of solutions to the following linear Schrödinger Kirchhoff Poisson equation with critical exponential growth on the full space □3 by using the discontinuous finite element (DG) as well as the principle of centralized compactness: <inline-formula>
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<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mo>{</mo> <mrow><mtable><mtr><mtd><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mrow><msub><mo>∫</mo><mrow><msup><mrow><mi>□</mi></mrow><mn>3</mn></msup></mrow></msub><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mo>∇</mo><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow></mrow></mrow><mo>)</mo></mrow><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>+</mo><mi>φ</mi><mi>u</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>u</mi><mi mathvariant="normal">Δ</mi><mrow><mo>(</mo><mrow><msup><mrow><mi>u</mi></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>=</mo><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><msup><mrow><mi>□</mi></mrow><mn>3</mn></msup></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>φ</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mn>2</mn></msup><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><msup><mrow><mi>□</mi></mrow><mn>3</mn></msup></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{{ - \left( {a + b\int_{{\square^3}} {{{\left| {\nabla u} \right|}^2}} } \right)\Delta u + V\left( x \right)u + \phi u - {1 \over 2}u\Delta \left( {{u^2}} \right) = K\left( x \right){u^{p - 2}}u,} &amp; {x \in {\square^3}} \cr { - \Delta \phi = {u^2},} &amp; {x \in {\square^3}} \cr } } \right.</tex-math>
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</inline-formula>, <italic>x</italic> ∈. Secondly, we make reasonable assumptions on the <italic>V</italic>, <italic>K</italic> , <italic>f</italic> functions of the equation, and use the principle of variational division to firstly obtain the corresponding energy generalization of this equation, and then we prove the corresponding generalizations of the equation satisfy the (<italic>C</italic>)<italic>c</italic> conditions. Finally, the existence of the solution of the equation is obtained by numerical simulation and then by using the Yamaji Lemma. The results show that the error of the finite element solution of the linear Schrödinger Kirchhoff Poisson equation in the spatial direction P1 reaches the optimal estimation under the <italic>L</italic>2 -parameter in an intermittent finite element numerical simulation environment, i.e., it is proved that there exist at least 1 and 3 positive solutions to the problem. The paper achieves rich research results which are informative for the solution of this class of linear differential equations.
</abstract>

In this paper, we firstly explore the existence of solutions to the following linear Schrödinger Kirchhoff Poisson equation with critical exponential growth on the full space □3 by using the discontinuous finite element (DG) as well as the principle of centralized compactness: 


{ −(a+b∫□3| ∇u |2)Δu+V(x)u+φu-12uΔ(u2)=K(x)up−2u,x∈□3−Δφ=u2,x∈□3 
\left\{ {\matrix{{ - \left( {a + b\int_{{\square^3}} {{{\left| {\nabla u} \right|}^2}} } \right)\Delta u + V\left( x \right)u + \phi u - {1 \over 2}u\Delta \left( {{u^2}} \right) = K\left( x \right){u^{p - 2}}u,} & {x \in {\square^3}}  \cr { - \Delta \phi  = {u^2},} & {x \in {\square^3}}  \cr  } } \right.

, x ∈. Secondly, we make reasonable assumptions on the V, K , f functions of the equation, and use the principle of variational division to firstly obtain the corresponding energy generalization of this equation, and then we prove the corresponding generalizations of the equation satisfy the (C)c conditions. Finally, the existence of the solution of the equation is obtained by numerical simulation and then by using the Yamaji Lemma. The results show that the error of the finite element solution of the linear Schrödinger Kirchhoff Poisson equation in the spatial direction P1 reaches the optimal estimation under the L2 -parameter in an intermittent finite element numerical simulation environment, i.e., it is proved that there exist at least 1 and 3 positive solutions to the problem. The paper achieves rich research results which are informative for the solution of this class of linear differential equations.

Existence analysis of solutions to the linear Schrödinger Kirchhoff Poisson equation based on interrupted finite elements

Applied Mathematics and Nonlinear Sciences

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

In this paper, we firstly explore the existence of solutions to the following linear Schrödinger Kirchhoff Poisson equation with critical exponential growth on...

Existence analysis of solutions to the linear Schrödinger Kirchhoff Poisson equation based on interrupted finite elements

Published Online: Jan 31, 2024

Page range: -

Received: Dec 15, 2023

Accepted: Dec 23, 2023

DOI: https://doi.org/10.2478/amns-2024-0104

Keywords
Interrupted finite elements, Schrödinger’s Kirchhoff Poisson equation, Variational principle, Numerical simulation, Existence

© 2024 Yafei Chen et al., published by Sciendo

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

Existence analysis of solutions to the linear Schrödinger Kirchhoff Poisson equation based on interrupted finite elements

Published Online: Jan 31, 2024

Page range: -

Received: Dec 15, 2023

Accepted: Dec 23, 2023

DOI: https://doi.org/10.2478/amns-2024-0104

KeywordsInterrupted finite elements, Schrödinger’s Kirchhoff Poisson equation, Variational principle, Numerical simulation, Existence

© 2024 Yafei Chen et al., published by Sciendo

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

Keywords
Interrupted finite elements, Schrödinger’s Kirchhoff Poisson equation, Variational principle, Numerical simulation, Existence