Positive solutions to an elliptic equation in ℝ^N of the Kirchhoff type

2016/03/24 by Yisheng Huang, Huang, Yisheng, Zeng Liu +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1603.07428

openalex publication_date 2016/03/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the following Kirchhoff type problem \\alignedamp;-\biggl(a + b∫_ℝ^N |∇ u|² dx \biggr) Δu + V(x) u = |u|^p-2u amp; in ℝ^N,\cr amp;u∈ H¹(ℝ^N), \endaligned. \eqno(P_a,b) where N≥3, 20 are parameters and V(x) is a potential function. Under some mild conditions on V(x), we prove that (P_a,b) has a positive solution for b small enough by the variational method, a non-existence result is also established in the cases N≥4. Our results in the case N=3 partial improve the results in \citeG15,LY14 and our results in the cases N≥4 are totally new to the best of our knowledge. By combining the scaling technique, we also give a global description on the structure of the positive solutions to the autonomous form of (P_a,b), that is V(x)≡λ>0. This result can be seen as a partial complement of the studies in \citeA12,A13.

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• On a critical Kirchhoff problem in high dimensions

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