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

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Abstract

This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schrödinger-type equations of the form

$$\begin{aligned} \left\{ \begin{array}{ll} - \displaystyle \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x)|u|^{P^+_+-2}u =\lambda f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array}\right. \end{aligned}$$

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.

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Acknowledgements

The authors thank a knowledgeable anonymous reviewer for very useful comments and remarks, which considerably improved the preliminary version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

    G. A. Afrouzi

  2. Department of Mathematics, Farhangian University, Tehran, Iran

    M. Mirzapour

  3. Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059, Kraków, Poland

    Vicenţiu D. Rădulescu

  4. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764 , 014700, Bucharest, Romania

    Vicenţiu D. Rădulescu

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  1. G. A. Afrouzi
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  2. M. Mirzapour
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  3. Vicenţiu D. Rădulescu
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Correspondence to Vicenţiu D. Rădulescu.

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Afrouzi, G.A., Mirzapour, M. & Rădulescu, V.D. Variational analysis of anisotropic Schrödinger equations without Ambrosetti–Rabinowitz-type condition. Z. Angew. Math. Phys. 69, 9 (2018). https://doi.org/10.1007/s00033-017-0900-y

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