# Variational analysis of anisotropic Schrödinger equations without Ambrosetti–Rabinowitz-type condition

Variational analysis of anisotropic Schrödinger equations without Ambrosetti–Rabinowitz-type... This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schrödinger-type equations of the form \begin{aligned} \left\{ \begin{array}{ll} - \displaystyle \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x)|u|^{P^+_+-2}u =\lambda f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array}\right. \end{aligned} - ∑ i = 1 N ∂ x i a i ( x , ∂ x i u ) + b ( x ) | u | P + + - 2 u = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , without the Ambrosetti–Rabinowitz growth condition. Our arguments rely on the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Zeitschrift für angewandte Mathematik und Physik Springer Journals

# Variational analysis of anisotropic Schrödinger equations without Ambrosetti–Rabinowitz-type condition

, Volume 69 (1) – Dec 14, 2017
17 pages

/lp/springer_journal/variational-analysis-of-anisotropic-schr-dinger-equations-without-ojvhl6H1Db
Publisher
Springer International Publishing
Copyright © 2017 by Springer International Publishing AG, part of Springer Nature
Subject
Engineering; Theoretical and Applied Mechanics; Mathematical Methods in Physics
ISSN
0044-2275
eISSN
1420-9039
D.O.I.
10.1007/s00033-017-0900-y
Publisher site
See Article on Publisher Site

### Abstract

This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schrödinger-type equations of the form \begin{aligned} \left\{ \begin{array}{ll} - \displaystyle \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x)|u|^{P^+_+-2}u =\lambda f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array}\right. \end{aligned} - ∑ i = 1 N ∂ x i a i ( x , ∂ x i u ) + b ( x ) | u | P + + - 2 u = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , without the Ambrosetti–Rabinowitz growth condition. Our arguments rely on the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter.

### Journal

Zeitschrift für angewandte Mathematik und PhysikSpringer Journals

Published: Dec 14, 2017

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