Abstract

We describe a result on the asymptotic behavior of the solutions of a system with two elliptic equations in RN\mathbb{R}^{N} involving a small parameter. More precisely, we study the system {−ε2div(a(x)∇u)+u=Qu(u,v)+γ2∗Ku(u,v)  in RN,−ε2Δv+b(x)v=Qv(u,v)+γ2∗Kv(u,v)  in RN,u,v∈H1(RN), u(x), v(x)>0  for each x∈RN,\left\{ \begin{aligned} -\varepsilon^{2} \text{div}(a(x) \nabla u)+u & = Q_{u}(u,v)+\frac{\gamma}{2^*} K_u(u,v)\ \ \text{in } \mathbb{R}^N, \\ -\varepsilon^{2} \Delta v + b(x) v & = Q_{v}(u,v)+\frac{\gamma}{2^*} K_v(u,v)\ \ \text{in } \mathbb{R}^N, \\ u,v \in H^{1}(\mathbb{R}^N), &\ u(x),\ v(x)>0\ \ \text{for each } x \in\mathbb{R}^N, \end{aligned} \right. where 2∗=2N/(N−2)2^*=2N/(N-2), N≥3N\geq 3, ε>0\varepsilon>0, aa and bb are positive continuous potentials, and QQ and KK are homogeneous functions with KK having critical growth. We use the penalization method for system introduced by C. O. Alves [Local mountain pass for a class of elliptic system, J. Math. Analysis Appl. 335 (2007) 135–150] in order to find a family of solutions (uε,vε)(u_{\varepsilon}, v_{\varepsilon}) in H1(RN)×H1(RN)H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N) such that, if Πε,a\Pi_{\varepsilon,a} and Πε,b\Pi_{\varepsilon, b} are maximum points of uεu_{\varepsilon} and vεv_{\varepsilon} respectively, then lim⁡ε→0+a(Πε,a)=inf⁡x∈RNa(x)   and  lim⁡ε→0+b(Πε,b)=inf⁡x∈RNb(x).\lim_{\varepsilon \rightarrow 0^+}a(\Pi_{\varepsilon, a}) = \inf_{x \in \mathbb{R}^{N}} a(x) \ \ \ \text{and}\ \ \lim_{\varepsilon \rightarrow 0^+}b(\Pi_{\varepsilon, b})= \displaystyle\inf_{x \in \mathbb{R}^{N}} b(x). Moreover, we relate the number of solutions with the topology of the set where the potentials aa and bb attain their minima. We consider the subcritical case γ=0\gamma=0 and the critical case γ=1\gamma=1.

Author information

Contact details are reproduced from the original publication and may be historical.

Segundo M. A. Salirrosas

Universidade de Brasília, Dep. de Matemática, Brasília, Brazil

semaarsa@gmail.com

Keywords

  • Elliptic systems
  • Schrödinger equation
  • Ljusternik-Schnirelmann theory
  • positive solutions

Mathematics Subject Classification

35J20, 35J50, 58E05

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Suggested citation

S. M. A. Salirrosas. “On Concentration Behavior and Multiplicity of Solutions for a System in ℝ^N.” Journal of Convex Analysis 30 (2023), No. 1, 175–204. https://doi.org/10.68381/jca30011

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