Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth. (February 2016)

• Record Type: Journal Article
• Authors: Shi, Hongxia; Chen, Haibo
• Is Part Of: Computers & mathematics with applications. Volume 71:issue 3(2016)
• Page Start: 849
• Page End: 858
• Publication Date: 2016-02
• Subjects: Mountain Pass Theorem -- Asymptotically periodic -- Critical growth -- Concentration-compactness principle; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2016.01.007
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: This paper is concerned with the following quasilinear Schrödinger equations: { − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = λ f ( x, u ) + g ( u ) | G ( u ) | 2 ∗ − 2 G ( u ), x ∈ R N, u ( x ) > 0, x ∈ R N, where N ≥ 3, 2 ∗ = 2 N N − 2, G ( u ) = ∫ 0 u g ( t ) d t and λ > 0 is a parameter. By using a change of variable, the quasilinear equation is reduced to a semilinear one, whose associated functional is well defined in H 1 ( R N ) . We establish the existence of positive solutions for this problem by using the Mountain Pass Theorem in combination with the concentration-compactness principle under appropriate assumptions on V ( x ) and f ( x, u ) . Recent results from the literature are improved and extended.