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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

Yinbin Deng 1, , Yi Li 2, and  Wei Shuai 3,

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.
Keywords: p-Laplacain type equations, weighted Sobolev space, critical growth, variational method., vanishing potential.
Mathematics Subject Classification: 35J60, 35A1.
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

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