Multiple solutions for a singular quasilinear elliptic system

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.