Journal
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
Volume 72, Issue 1, Pages -Publisher
SPRINGER INT PUBL AG
DOI: 10.1007/s00033-020-01455-w
Keywords
Kirchhoff problem; Critical exponential growth; Trudinger-Moser inequality; Nehari-type ground-state solution
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Funding
- National Natural Science Foundation of China [11971485, 12001542]
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This paper proves the existence of nontrivial solutions and Nehari-type ground-state solutions for a Kirchhoff-type elliptic equation. It introduces new approaches to precisely estimate the minimax level of the energy functional and extends previous results by providing more accurate estimations and a simple proof of a known inequality.
In this paper, we prove the existence of nontrivial solutions and Nehari-type ground-state solutions for the following Kirchhoff-type elliptic equation: {-m(parallel to del u parallel to(2)(2))Delta u = f(x, u), in Omega, u = 0, on partial derivative Omega, where Omega subset of R-2 is a smooth bounded domain, m : R+ -> R+ is a Kirchhoff function, and f has critical exponential growth in the sense of Trudinger-Moser inequality. We develop some new approaches to estimate precisely the minimax level of the energy functional and prove the existence of Nehari-type ground-state solutions and nontrivial solutions for the above problem. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level, and also give a simple proof of a known inequality due to P.L. Lions.
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