Existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems

Abstract

Using the minimax methods in critical point theory and a generalized Landesman-Lazer type condition, we establish two existence results of solutions for asymptotically linear noncooperative elliptic systems at resonance. Besides this, we obtain two solutions in the case of near resonance.

Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

“…which have the same meaning as T . Next, we consider the existence results of system (S λ ,••• ,n ) by using a completely di erent method from that of [2,8,10,11,[14][15][16]20,21,29,30,32,35,36,40,41,46,49,50,[53][54][55]61], namely the eigenvalue theory:…”

“…At the same time, we notice that many authors have paid more attention to various of system problems, for example, see [2,10,[14][15][16]20,21,29,30,32,35,40,46,49,50]. Specially, Lair and Wood [36] analyzed the existence of entire positive solutions of system ∆u = p(|x|)v α , ∆v = q(|x|)u β ,…”

Multiplicity of Solutions for a Class of Noncooperative Elliptic Systems

“…In the previous works, the authors always need to make the assumptions on the nonlinearity at infinity. For example, the sub-quadratic cases were considered in [1], the superquadratic cases were considered in [2] [4] [7] [8] [12] [14] and the asymptotically quadratic cases were considered in [6] [9] [11] [13] [15] [16] [17]. Compared with these results, our nonlinearity F is general at infinity, we do not need to assume the behavior of the nonlinear term F at infinity.…”

Multiplicity result for asymptotically linear noncooperative elliptic systems

“…In , by using the fountain theorem due to and an abstract critical point theorem due to , Zou obtained the multiplicity result under the condition that the potential is even. In , Ke and Tang used the minimax methods and a generalized Landesman–Lazer type condition to obtain the existence and multiplicity of nontrivial solutions.…”