caa a22 4500 606210016 CHVBK 20210128101031.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00032-015-0236-z doi (NATIONALLICENCE)springer-10.1007/s00032-015-0236-z Existence of Multi-bump Solutions for a Class of Quasilinear Schrödinger Equations in $${\mathbb{R}^{N}}$$ R N Involving Critical Growth [Elektronische Daten] [Sihua Liang, Jihui Zhang] In this paper we prove the existence of multi-bump solutions for a class of quasilinear Schrödinger equations of the form $${-\Delta{u} + (\lambda{V} (x) + Z(x))u - \Delta(u^{2})u = \beta{h}(u) + u^{22*-1}}$$ - Δ u + ( λ V ( x ) + Z ( x ) ) u - Δ ( u 2 ) u = β h ( u ) + u 22 ∗ - 1 in the whole space, where h is a continuous function, $${V, Z : \mathbb{R}^{N} \rightarrow \mathbb{R}}$$ V , Z : R N → R are continuous functions. We assume that V(x) is nonnegative and has a potential well $${\Omega : = {\rm int} V^{-1}(0)}$$ Ω : = int V - 1 ( 0 ) consisting of k components $${\Omega_{1}, \ldots , \Omega{k}}$$ Ω 1 , ... , Ω k such that the interior of Ω i is not empty and $${\partial\Omega_{i}}$$ ∂ Ω i is smooth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that for any given non-empty subset. $${\Gamma \subset \{1, \ldots ,k\}}$$ Γ ⊂ { 1 , ... , k } , a bump solution is trapped in a neighborhood of $${\cup_{{j}\in\Gamma}\Omega_{j}}$$ ∪ j ∈ Γ Ω j for $${\lambda > 0}$$ λ > 0 large enough. Springer Basel, 2015 Quasilinear Schrödinger equation nationallicence critical growth nationallicence multi-bump solutions nationallicence variational methods nationallicence Liang Sihua College of Mathematics, Changchun Normal University, 130032, Changchun, Jilin, P.R. China aut Zhang Jihui Key Laboratory of Symbolic Computation, and Knowledge Engineering of Ministry of Education, Jilin University, 130012, Changchun, P.R. China aut Milan Journal of Mathematics Springer Basel 83/1(2015-06-01), 55-90 1424-9286 83:1<55 2015 83 32 https://doi.org/10.1007/s00032-015-0236-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00032-015-0236-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Liang Sihua College of Mathematics, Changchun Normal University, 130032, Changchun, Jilin, P.R. China aut NATIONALLICENCE 700 1- Zhang Jihui Key Laboratory of Symbolic Computation, and Knowledge Engineering of Ministry of Education, Jilin University, 130012, Changchun, P.R. China aut NATIONALLICENCE 773 0- Milan Journal of Mathematics Springer Basel 83/1(2015-06-01), 55-90 1424-9286 83:1<55 2015 83 32