EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO A CLASS OF NONCOOPERATIVE ELLIPTIC SYSTEMS WITH SUPERLINEAR NONLINEAR TERMS

2019 ◽  
Vol 9 (4) ◽  
pp. 1347-1358
Author(s):  
Xiaofeng Ke ◽  
◽  
Chunlei Tang
Keyword(s):  
Elliptic Systems ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

scholarly journals On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

2020 ◽  
Vol 9 (1) ◽  
pp. 1480-1503
Author(s):  
Mousomi Bhakta ◽  
Phuoc-Tai Nguyen
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Total Mass ◽  
Elliptic Systems ◽  
Linking Theorem ◽  
Bounded Domains ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Smallness Condition

Abstract We study positive solutions to the fractional Lane-Emden system $$\begin{array}{} \displaystyle \left\{ \begin{aligned} (-{\it\Delta})^s u &= v^p+\mu \quad &\text{in } {\it\Omega} \\(-{\it\Delta})^s v &= u^q+\nu \quad &\text{in } {\it\Omega}\\u = v &= 0 \quad &&\!\!\!\!\!\!\!\!\!\!\!\!\text{in } {\it\Omega}^c={\mathbb R}^N \setminus {\it\Omega}, \end{aligned} \right. \end{array}$$(S) where Ω is a C2 bounded domains in ℝN, s ∈ (0, 1), N > 2s, p > 0, q > 0 and μ, ν are positive measures in Ω. We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of μ and ν. Furthermore, if p, q ∈ $\begin{array}{} (1,\frac{N+s}{N-s}) \end{array}$ and 0 ≤ μ, ν ∈ Lr(Ω), for some r > $\begin{array}{} \frac{N}{2s}, \end{array}$ we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.

scholarly journals Existence and multiplicity of solutions for a class of superlinear elliptic systems

2018 ◽  
Vol 7 (2) ◽  
pp. 183-196 ◽  
Author(s):  
Chun Li ◽  
Ravi P. Agarwal ◽  
Dong-Lun Wu
Keyword(s):  
Critical Point ◽  
Growth Condition ◽  
Critical Point Theory ◽  
Elliptic Systems ◽  
Point Theory ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Minimax Methods

AbstractIn this paper, we establish the existence and multiplicity of solutions for a class of superlinear elliptic systems without Ambrosetti and Rabinowitz growth condition. Our results are based on minimax methods in critical point theory.

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