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

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 1977-1991. doi: 10.1016/j.jde.2012.11.013.
[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), 117-144. doi: 10.4171/JEMS/24.
[3]	A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications, J. Funct. Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.
[4]	A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Integ. Eqns., 18 (2005), 1321-1332.
[5]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.
[6]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[7]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure. Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[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), 273-301. doi: 10.1007/s10231-009-0109-6.
[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), 1157-1178. doi: 10.1017/S0308210506000795.
[10]	Y. B. Deng, The existence and nodal character of the solutions in $\mathbb{R}^N2$ for semilinear elliptic equation involving critical Sobolev exponent, Acta Math. Sci., 9 (1989), 385-402.
[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), 1376-1398. doi: 10.1016/j.jde.2012.05.009.
[12]	Y. B. Deng, Z. H. Guo and G. S. Wang, Nodal solutions for $p-$Laplace equations with critical growth, Nonlin. Analysis, TMA., 54 (2003), 1121-1151. doi: 10.1016/S0362-546X(03)00129-9.
[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), 197-229. doi: 10.1006/jdeq.2001.4077.
[14]	M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of heat equations, Nonlin Analysis, TMA, 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0.
[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), 2732-2743. doi: 10.1016/j.jde.2013.01.007.
[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), 64-74.
[17]	P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part I, Rev. Mat. Iber., 1 (1985), 145-201. doi: 10.4171/RMI/6.
[18]	P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part II, Rev. Mat. Iber., 1 (1985), 45-121. doi: 10.4171/RMI/12.
[19]	B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, 1990.
[20]	J. Serrin, Local behavior of solutions of quasilinear equations, Acta. Math., 111 (1964), 247-302. doi: 10.1007/BF02391014.
[21]	D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938. doi: 10.1090/S0002-9947-04-03769-9.
[22]	W. A. Strauss, Existence of solitary in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.
[23]	C. A. Swanson and L. S. Yu., Critical p-laplacian problems in $\mathbb{R}^N2$, Annali Mat. Pura Appl., 169 (1995), 233-250. doi: 10.1007/BF01759355.
[24]	P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqns., 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[25]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.
[26]	X. P. Zhu, Nontrivial solution of quasilinear elliptic equation involving critical Sobolev exponent, Sci. Sinica., 31 (1990), 1161-1181.

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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), 1977-1991. doi: 10.1016/j.jde.2012.11.013.
[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), 117-144. doi: 10.4171/JEMS/24.
[3]	A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications, J. Funct. Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.
[4]	A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Diff. Integ. Eqns., 18 (2005), 1321-1332.
[5]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.
[6]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[7]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure. Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[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), 273-301. doi: 10.1007/s10231-009-0109-6.
[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), 1157-1178. doi: 10.1017/S0308210506000795.
[10]	Y. B. Deng, The existence and nodal character of the solutions in $\mathbb{R}^N2$ for semilinear elliptic equation involving critical Sobolev exponent, Acta Math. Sci., 9 (1989), 385-402.
[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), 1376-1398. doi: 10.1016/j.jde.2012.05.009.
[12]	Y. B. Deng, Z. H. Guo and G. S. Wang, Nodal solutions for $p-$Laplace equations with critical growth, Nonlin. Analysis, TMA., 54 (2003), 1121-1151. doi: 10.1016/S0362-546X(03)00129-9.
[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), 197-229. doi: 10.1006/jdeq.2001.4077.
[14]	M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of heat equations, Nonlin Analysis, TMA, 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0.
[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), 2732-2743. doi: 10.1016/j.jde.2013.01.007.
[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), 64-74.
[17]	P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part I, Rev. Mat. Iber., 1 (1985), 145-201. doi: 10.4171/RMI/6.
[18]	P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part II, Rev. Mat. Iber., 1 (1985), 45-121. doi: 10.4171/RMI/12.
[19]	B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, 219. Longman Scientific & Technical, Harlow, 1990.
[20]	J. Serrin, Local behavior of solutions of quasilinear equations, Acta. Math., 111 (1964), 247-302. doi: 10.1007/BF02391014.
[21]	D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938. doi: 10.1090/S0002-9947-04-03769-9.
[22]	W. A. Strauss, Existence of solitary in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.
[23]	C. A. Swanson and L. S. Yu., Critical p-laplacian problems in $\mathbb{R}^N2$, Annali Mat. Pura Appl., 169 (1995), 233-250. doi: 10.1007/BF01759355.
[24]	P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqns., 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[25]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.
[26]	X. P. Zhu, Nontrivial solution of quasilinear elliptic equation involving critical Sobolev exponent, Sci. Sinica., 31 (1990), 1161-1181.

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