American Institute of Mathematical Sciences

Advanced
March  2009, 8(2): 533-557. doi: 10.3934/cpaa.2009.8.533

Nodal solutions to critical growth elliptic problems under Steklov boundary conditions

Elvise Berchio 1, , Filippo Gazzola 2, and  Dario Pierotti 3,

1. 

Dipartimento SEMEQ, Università del Piemonte Orientale, via E. Perrone 18, Novara, 28100, Italy
2. 

Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano
3. 

Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 32, Milano, 20133, Italy

Received  January 2008 Revised  July 2008 Published  December 2008

We study elliptic problems at critical growth under Steklov boundary conditions in bounded domains. For a second order problem we prove existence of nontrivial nodal solutions. These are obtained by combining a suitable linking argument with fine estimates on the concentration of Sobolev minimizers on the boundary. When the domain is the unit ball, we obtain a multiplicity result by taking advantage of the explicit form of the Steklov eigenfunctions. We also partially extend the results in the ball to the case of fourth order Steklov boundary value problems.
Keywords: nodal solutions, Steklov, Critical growth.
Mathematics Subject Classification: Primary: 35J20, 35J35; Secondary: 35J40.
Citation: Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[1]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[2]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[3]

Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure & Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
[4]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[5]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921
[6]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
[7]

Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729
[8]

Nguyen Lam, Guozhen Lu. Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2187-2205. doi: 10.3934/dcds.2012.32.2187
[9]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921
[10]

Marcos L. M. Carvalho, José Valdo A. Goncalves, Claudiney Goulart, Olímpio H. Miyagaki. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 83-106. doi: 10.3934/cpaa.2019006
[11]

Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 809-823. doi: 10.3934/dcdss.2011.4.809
[12]

M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703
[13]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417
[14]

Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
[15]

Monica Lazzo, Paul G. Schmidt. Nodal properties of radial solutions for a class of polyharmonic equations. Conference Publications, 2007, 2007 (Special) : 634-643. doi: 10.3934/proc.2007.2007.634
[16]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393
[17]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273
[18]

Claudianor O. Alves, J. V. Gonçalves, Olimpio Hiroshi Miyagaki. Remarks on multiplicity of positive solutions of nonlinear elliptic equations in $IR^N$ with critical growth. Conference Publications, 1998, 1998 (Special) : 51-57. doi: 10.3934/proc.1998.1998.51
[19]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
[20]

Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences