Existence and Stability in Nonlocal Schrödinger–Poisson–Slater Equations
In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>Δ</mo><mi>u</mi><mo>...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/6/329 |
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| Summary: | In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>Δ</mo><mi>u</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>λ</mi><mfenced separators="" open="(" close=")"><msub><mi>I</mi><mi>α</mi></msub><mo>∗</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mi>q</mi></msup></mfenced><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>,</mo><mi>p</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>I</mi><mi>α</mi></msub></semantics></math></inline-formula> is the Riesz potential. We obtain the existence, stability, and symmetry-breaking of solutions for both radial and nonradial cases. In the radial case, we use variational methods to establish the coercivity and weak lower semicontinuity of the energy functional, ensuring the existence of a positive solution when <i>p</i> is below a critical threshold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>4</mn><mi>q</mi><mo>+</mo><mn>2</mn><mi>α</mi></mrow><mrow><mn>2</mn><mo>+</mo><mi>α</mi></mrow></mfrac></mstyle></mrow></semantics></math></inline-formula>. In addition, we prove that the energy functional attains a minimum, guaranteeing the existence of a ground-state solution under specific conditions on the parameters. We also apply the Pohozaev identity to identify parameter regimes where only the trivial solution is possible. In the nonradial case, we use the Nehari manifold method to prove the existence of ground-state solutions, analyze symmetry-breaking by studying the behavior of the energy functional and identifying the parameter regimes in the nonradial case, and apply concentration-compactness methods to prove the global well-posedness of the Cauchy problem and demonstrate the orbital stability of the ground state. Our results demonstrate the stability of solutions in both radial and nonradial cases, identifying critical parameter regimes for stability and instability. This work enhances our understanding of the role of nonlocal interactions in symmetry-breaking and stability, while extending existing theories to multiparameter and higher-dimensional settings in the Schrödinger–Poisson–Slater model. |
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| ISSN: | 2504-3110 |