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Multiple solutions for a singular quasilinear elliptic system.

Lin Chen1, Caisheng Chen, Zonghu Xiu

  • 1College of Science, Hohai University, Nanjing 210098, China ; College of Mathematics and Statistics, Yili Normal University, Yining 835000, China.

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Summary
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This study proves at least two nontrivial solutions for a quasilinear elliptic system using the Mountain Pass Theorem and Ekeland's variational principle. The research focuses on the multiplicity of solutions in a specific mathematical context.

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Area of Science:

  • Mathematical Analysis
  • Nonlinear Partial Differential Equations
  • Quasilinear Elliptic Systems

Background:

  • Investigates the existence and multiplicity of solutions for complex quasilinear elliptic systems.
  • Addresses systems with specific nonlinear terms and weighted p-Laplacians.
  • Considers solutions in Euclidean space ℝ(N) with positivity constraints.

Purpose of the Study:

  • To establish the existence of at least two nontrivial solutions for a given quasilinear elliptic system.
  • To analyze the multiplicity of solutions under specified parameter ranges and function conditions.
  • To apply advanced variational methods to a challenging class of nonlinear equations.

Main Methods:

  • Employs the Mountain Pass Theorem, a critical tool in critical point theory for finding critical points of functionals.
  • Utilizes Ekeland's variational principle to guarantee the existence of solutions.
  • Involves detailed analysis of the functional associated with the quasilinear elliptic system, including compactness conditions and growth properties.

Main Results:

  • Demonstrates that the considered quasilinear elliptic system possesses at least two nontrivial solutions.
  • The results hold for a specific range of parameters (λ, μ > 0) and exponents (1 < p < N, 1 < γ < p < α + β < p*).
  • Confirms the applicability of variational methods to problems involving weighted p-Laplacians and mixed nonlinearity terms.

Conclusions:

  • The study successfully proves the existence of multiple nontrivial solutions for the investigated quasilinear elliptic system.
  • The findings contribute to the understanding of solution multiplicity in nonlinear elliptic equations.
  • The applied methods provide a framework for analyzing similar nonlinear systems in mathematical physics and analysis.