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Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials
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 |
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. |
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), 1977-1991.
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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