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n-Kirchhoff type equations with exponential nonlinearities

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this article, we study the existence of non-negative solutions of the class of non-local problem of n-Kirchhoff type

$$\begin{aligned} \left\{ \begin{array}{l} -m(\int _{\Omega }|\nabla u|^n)\Delta _n u = f(x,u) \; \text {in}\; \Omega ,\quad u =0\quad \text {on} \quad \partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \subset \mathbb R^n\) is a bounded domain with smooth boundary, \(n\ge 2\) and f behaves like \(e^{|u|^{\frac{n}{n-1}}}\) as \(|u|\rightarrow \infty \). Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when f(x, t) is concave near \(t=0\) and convex as \(t\rightarrow \infty \).

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Acknowledgments

Third author’s research is supported by National Board for Higher Mathematics, Govt. of India, Grant Number: 2/48(12)/2012/NBHM(R.P.)/R & D II/13095.

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  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110 016, India

    Sarika Goyal, Pawan Kumar Mishra & K. Sreenadh

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  1. Sarika Goyal
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  2. Pawan Kumar Mishra
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  3. K. Sreenadh
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Correspondence to K. Sreenadh.

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Goyal, S., Mishra, P.K. & Sreenadh, K. n-Kirchhoff type equations with exponential nonlinearities. RACSAM 110, 219–245 (2016). https://doi.org/10.1007/s13398-015-0230-x

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  • DOI: https://doi.org/10.1007/s13398-015-0230-x

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