Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

IF 0.7 2区 数学 Q2 MATHEMATICS
Rachid Echarghaoui, Rachid Sersif, Zakaria Zaimi
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引用次数: 0

Abstract

The authors of Cao and Yan (J Differ Equ 251:1389–1414, 2011) have considered the following semilinear critical Neumann problem

$$\begin{aligned} \varvec{-\Delta u=\vert u\vert ^{2^{*}-2} u+g(u) \quad \text{ in } \Omega , \quad \frac{\partial u}{\partial \nu }=0 \quad \text{ on } \partial \Omega ,} \end{aligned}$$

where \(\varvec{\Omega }\) is a bounded domain in \(\varvec{\mathbb {R}^{N}}\) satisfying some geometric conditions, \(\varvec{\nu }\) is the outward unit normal of \(\varvec{\partial \Omega , 2^{*}:=\frac{2 N}{N-2}}\) and \(\varvec{g(t):=\mu \vert t\vert ^{p-2} t-t,}\) where \(\varvec{p \in \left( 2,2^{*}\right) }\) and \(\varvec{\mu >0}\) are constants. They proved the existence of infinitely many solutions with positive energy for the above problem if \(\varvec{N>\max \left( \frac{2(p+1)}{p-1}, 4\right) .}\) In this present paper, we consider the case where the exponent \(\varvec{p \in \left( 1,2\right) }\) and we show that if \(\varvec{N>\frac{2(p+1)}{p-1},}\) then the above problem admits an infinite set of solutions with positive energy. Our main result extend that obtained by P. Han in [9] for the case of elliptic problem with Dirichlet boundary conditions.

具有临界Sobolev指数和凹凸非线性的半线性Neumann方程的无穷多正能量解
Cao和Yan的作者(J Differ Equ 251:1389-1414, 2011)考虑了以下半线性临界诺伊曼问题 $$\begin{aligned} \varvec{-\Delta u=\vert u\vert ^{2^{*}-2} u+g(u) \quad \text{ in } \Omega , \quad \frac{\partial u}{\partial \nu }=0 \quad \text{ on } \partial \Omega ,} \end{aligned}$$在哪里 \(\varvec{\Omega }\) 有界域在吗 \(\varvec{\mathbb {R}^{N}}\) 满足一些几何条件, \(\varvec{\nu }\) 向外单位是法向的吗 \(\varvec{\partial \Omega , 2^{*}:=\frac{2 N}{N-2}}\) 和 \(\varvec{g(t):=\mu \vert t\vert ^{p-2} t-t,}\) 在哪里 \(\varvec{p \in \left( 2,2^{*}\right) }\) 和 \(\varvec{\mu >0}\) 都是常数。证明了上述问题存在无穷多个正能量解 \(\varvec{N>\max \left( \frac{2(p+1)}{p-1}, 4\right) .}\) 在本文中,我们考虑指数 \(\varvec{p \in \left( 1,2\right) }\) 我们证明了 \(\varvec{N>\frac{2(p+1)}{p-1},}\) 那么上述问题就有无限多的正能量解。对于具有Dirichlet边界条件的椭圆型问题,我们的主要结果推广了P. Han[9]所得到的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Collectanea Mathematica
Collectanea Mathematica 数学-数学
CiteScore
2.70
自引率
9.10%
发文量
36
审稿时长
>12 weeks
期刊介绍: Collectanea Mathematica publishes original research peer reviewed papers of high quality in all fields of pure and applied mathematics. It is an international journal of the University of Barcelona and the oldest mathematical journal in Spain. It was founded in 1948 by José M. Orts. Previously self-published by the Institut de Matemàtica (IMUB) of the Universitat de Barcelona, as of 2011 it is published by Springer.
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