• Previous Article
    Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials
  • DCDS Home
  • This Issue
  • Next Article
    The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds
February  2016, 36(2): 683-699. doi: 10.3934/dcds.2016.36.683

Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity

1. 

Department of Mathematics, Huazhong Normal University, Wuhan 430079

2. 

Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA, United States

3. 

Department of Mathematics, Huazhong Normal University, Wuhan, 430079

Received  July 2014 Revised  February 2015 Published  August 2015

In this paper, we study the existence of positive solution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V (x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*} where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac {Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded, nonnegative continuous function. By working in the weighted Sobolev spaces, and using variational method, we prove that the given problem has at least one positive solution.
Citation: 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
References:
[1]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Diff. Eqns., 254 (2013), 1977. doi: 10.1016/j.jde.2012.11.013. Google Scholar

[2]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Eur. Math. Soc., 7 (2005), 117. doi: 10.4171/JEMS/24. Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications,, J. Funct. Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar

[4]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, Diff. Integ. Eqns., 18 (2005), 1321. Google Scholar

[5]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3. Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Ration. Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar

[7]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Commun. Pure. Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

[8]

D. Bonheure and J. Van Schaftingen, Ground states for the nonlinear Schrödinger equation with potentials vanishing at infinitly,, Annali Mat. Pura Appl., 189 (2010), 273. doi: 10.1007/s10231-009-0109-6. Google Scholar

[9]

F. Catrina, M. Furtado and M. Montenegro, Positive solutions for nonlinear elliptic equations with fast increasing weights,, Proc. Roy. Soc. Edinb. A, 137 (2007), 1157. doi: 10.1017/S0308210506000795. Google Scholar

[10]

Y. B. Deng, The existence and nodal character of the solutions in $\mathbbR^N$ for semilinear elliptic equation involving critical Sobolev exponent,, Acta Math. Sci., 9 (1989), 385. Google Scholar

[11]

Y. B. Deng, L. Y. Jin and S. J. Peng, Solutions of Schrodinger Equations with Inverse Square Potential and Critical Nonlinearity,, J. Diff. Eqns., 253 (2012), 1376. doi: 10.1016/j.jde.2012.05.009. Google Scholar

[12]

Y. B. Deng, Z. H. Guo and G. S. Wang, Nodal solutions for $p-$Laplace equations with critical growth,, Nonlin. Analysis, 54 (2003), 1121. doi: 10.1016/S0362-546X(03)00129-9. Google Scholar

[13]

Y. B. Deng and Y. Li, On the existence of multiple positive solutions for a semilinear problem in exterior domains,, J. Diff. Eqns., 181 (2002), 197. doi: 10.1006/jdeq.2001.4077. Google Scholar

[14]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of heat equations,, Nonlin Analysis, 11 (1987), 1103. doi: 10.1016/0362-546X(87)90001-0. Google Scholar

[15]

M. F. Furtado, J. P. P. da Silva and M. S. Xavier, Multiplicity of self-similar solutions for a critical equation,, J. Diff. Eqns., 254 (2013), 2732. doi: 10.1016/j.jde.2013.01.007. Google Scholar

[16]

G. B. Li, The existence of a weak solution of quasilinear elliptic equation with critical Sobolev exponent on unbounded domain,, Acta Math. Sci., 14 (1994), 64. Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part I,, Rev. Mat. Iber., 1 (1985), 145. doi: 10.4171/RMI/6. Google Scholar

[18]

P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part II,, Rev. Mat. Iber., 1 (1985), 45. doi: 10.4171/RMI/12. Google Scholar

[19]

B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Research Notes in Mathematics Series, (1990). Google Scholar

[20]

J. Serrin, Local behavior of solutions of quasilinear equations,, Acta. Math., 111 (1964), 247. doi: 10.1007/BF02391014. Google Scholar

[21]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities,, Trans. Amer. Math. Soc., 357 (2005), 2909. doi: 10.1090/S0002-9947-04-03769-9. Google Scholar

[22]

W. A. Strauss, Existence of solitary in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar

[23]

C. A. Swanson and L. S. Yu., Critical p-laplacian problems in $\mathbbR^N$,, Annali Mat. Pura Appl., 169 (1995), 233. doi: 10.1007/BF01759355. Google Scholar

[24]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Diff. Eqns., 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar

[25]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[26]

X. P. Zhu, Nontrivial solution of quasilinear elliptic equation involving critical Sobolev exponent,, Sci. Sinica., 31 (1990), 1161. Google Scholar

show all references

References:
[1]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Diff. Eqns., 254 (2013), 1977. doi: 10.1016/j.jde.2012.11.013. Google Scholar

[2]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Eur. Math. Soc., 7 (2005), 117. doi: 10.4171/JEMS/24. Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications,, J. Funct. Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar

[4]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, Diff. Integ. Eqns., 18 (2005), 1321. Google Scholar

[5]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.1090/S0002-9939-1983-0699419-3. Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Ration. Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar

[7]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Commun. Pure. Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

[8]

D. Bonheure and J. Van Schaftingen, Ground states for the nonlinear Schrödinger equation with potentials vanishing at infinitly,, Annali Mat. Pura Appl., 189 (2010), 273. doi: 10.1007/s10231-009-0109-6. Google Scholar

[9]

F. Catrina, M. Furtado and M. Montenegro, Positive solutions for nonlinear elliptic equations with fast increasing weights,, Proc. Roy. Soc. Edinb. A, 137 (2007), 1157. doi: 10.1017/S0308210506000795. Google Scholar

[10]

Y. B. Deng, The existence and nodal character of the solutions in $\mathbbR^N$ for semilinear elliptic equation involving critical Sobolev exponent,, Acta Math. Sci., 9 (1989), 385. Google Scholar

[11]

Y. B. Deng, L. Y. Jin and S. J. Peng, Solutions of Schrodinger Equations with Inverse Square Potential and Critical Nonlinearity,, J. Diff. Eqns., 253 (2012), 1376. doi: 10.1016/j.jde.2012.05.009. Google Scholar

[12]

Y. B. Deng, Z. H. Guo and G. S. Wang, Nodal solutions for $p-$Laplace equations with critical growth,, Nonlin. Analysis, 54 (2003), 1121. doi: 10.1016/S0362-546X(03)00129-9. Google Scholar

[13]

Y. B. Deng and Y. Li, On the existence of multiple positive solutions for a semilinear problem in exterior domains,, J. Diff. Eqns., 181 (2002), 197. doi: 10.1006/jdeq.2001.4077. Google Scholar

[14]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of heat equations,, Nonlin Analysis, 11 (1987), 1103. doi: 10.1016/0362-546X(87)90001-0. Google Scholar

[15]

M. F. Furtado, J. P. P. da Silva and M. S. Xavier, Multiplicity of self-similar solutions for a critical equation,, J. Diff. Eqns., 254 (2013), 2732. doi: 10.1016/j.jde.2013.01.007. Google Scholar

[16]

G. B. Li, The existence of a weak solution of quasilinear elliptic equation with critical Sobolev exponent on unbounded domain,, Acta Math. Sci., 14 (1994), 64. Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part I,, Rev. Mat. Iber., 1 (1985), 145. doi: 10.4171/RMI/6. Google Scholar

[18]

P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part II,, Rev. Mat. Iber., 1 (1985), 45. doi: 10.4171/RMI/12. Google Scholar

[19]

B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Research Notes in Mathematics Series, (1990). Google Scholar

[20]

J. Serrin, Local behavior of solutions of quasilinear equations,, Acta. Math., 111 (1964), 247. doi: 10.1007/BF02391014. Google Scholar

[21]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities,, Trans. Amer. Math. Soc., 357 (2005), 2909. doi: 10.1090/S0002-9947-04-03769-9. Google Scholar

[22]

W. A. Strauss, Existence of solitary in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar

[23]

C. A. Swanson and L. S. Yu., Critical p-laplacian problems in $\mathbbR^N$,, Annali Mat. Pura Appl., 169 (1995), 233. doi: 10.1007/BF01759355. Google Scholar

[24]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Diff. Eqns., 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar

[25]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[26]

X. P. Zhu, Nontrivial solution of quasilinear elliptic equation involving critical Sobolev exponent,, Sci. Sinica., 31 (1990), 1161. Google Scholar

[1]

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

[2]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215

[3]

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

[4]

Jiguang Bao, Nguyen Lam, Guozhen Lu. Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 577-600. doi: 10.3934/dcds.2016.36.577

[5]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079

[6]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691

[7]

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

[8]

Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124

[9]

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

[10]

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

[11]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[12]

Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008

[13]

Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096

[14]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365

[15]

Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021

[16]

Yaping Wu, Xiuxia Xing. Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1123-1139. doi: 10.3934/dcds.2008.20.1123

[17]

Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885

[18]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[19]

Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041

[20]

Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943

2018 Impact Factor: 1.143

Metrics

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

Other articles
by authors

[Back to Top]