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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. [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. [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. [4] A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, Diff. Integ. Eqns., 18 (2005), 1321. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Research Notes in Mathematics Series, (1990). [20] J. Serrin, Local behavior of solutions of quasilinear equations,, Acta. Math., 111 (1964), 247. doi: 10.1007/BF02391014. [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. [22] W. A. Strauss, Existence of solitary in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. [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. [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. [25] M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (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.

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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. 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. 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. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Research Notes in Mathematics Series, (1990). [20] J. Serrin, Local behavior of solutions of quasilinear equations,, Acta. Math., 111 (1964), 247. doi: 10.1007/BF02391014. [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. [22] W. A. Strauss, Existence of solitary in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. [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. [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. [25] M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (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.
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