March  2008, 7(2): 383-405. doi: 10.3934/cpaa.2008.7.383

On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function

1. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Received  September 2005 Revised  July 2007 Published  December 2007

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result to prove that the semilinear elliptic equation with a sign-changing weight function has at least two positive solutions.
Citation: Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383
[1]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076

[2]

Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285

[3]

M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113

[4]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108

[5]

Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107

[6]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036

[7]

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053

[8]

Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559

[9]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357

[10]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure and Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

[11]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715

[12]

João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621

[13]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231

[14]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[15]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857

[16]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427

[17]

Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008

[18]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[19]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875

[20]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (117)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]