October  2015, 35(10): 4889-4903. doi: 10.3934/dcds.2015.35.4889

Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms

1. 

Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

2. 

Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan

3. 

Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192, Japan

Received  October 2014 Revised  January 2015 Published  April 2015

The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights. Time local and global solutions are shown to exist in the energy class. The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.
Citation: Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889
References:
[1]

S. Adachi and K. Tanaka, A scale-invariant form of Trudinger-Moser inequality and its best exponent,, Proc. Amer. Math. Soc., 1102 (1999), 148.

[2]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.1090/S0894-0347-1989-1002633-4.

[3]

L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth,, Comm. Partial Differential Equations, 36 (2011), 797. doi: 10.1080/03605302.2010.534684.

[4]

A. A. Baraket, Local existence and estimations for a semilinear wave equation in two dimension space,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7 (2004), 1.

[5]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.1090/S0002-9947-1984-0742415-3.

[6]

C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988).

[7]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, (1976).

[8]

M. Bouchekif and A. Matallah, Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev-Hardy exponent,, Appl. Math. Lett., 22 (2009), 268. doi: 10.1016/j.aml.2008.03.024.

[9]

P. Brenner, $L_p$-estimates of difference schemes for strictly hyperbolic systems with nonsmooth data,, SIAM J. Numer. Anal., 14 (1977), 1126. doi: 10.1137/0714078.

[10]

H. Brézis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential,, Selecta Math. (N.S.), 11 (2005), 1. doi: 10.1007/s00029-005-0003-z.

[11]

H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: 10.1016/0362-546X(80)90068-1.

[12]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Partial Differential Equations, 5 (1980), 773. doi: 10.1080/03605308008820154.

[13]

N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential,, J. Funct. Anal., 203 (2003), 519. doi: 10.1016/S0022-1236(03)00238-6.

[14]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259.

[15]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 1267. doi: 10.3934/cpaa.2014.13.1267.

[16]

J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ., 6 (2009), 549. doi: 10.1142/S0219891609001927.

[17]

Z. Gan, Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential,, Commun. Pure Appl. Anal., 8 (2009), 1541. doi: 10.3934/cpaa.2009.8.1541.

[18]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 441. doi: 10.1006/jdeq.1997.3375.

[19]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation. II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15.

[20]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119.

[21]

T-S. Hsu and H-L. Lin, Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev-Hardy exponents,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 617. doi: 10.1017/S0308210509000729.

[22]

S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity,, Comm. Pure Appl. Math., 59 (2006), 1639. doi: 10.1002/cpa.20127.

[23]

S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant,, Proc. Amer. Math. Soc., 135 (2007), 87. doi: 10.1090/S0002-9939-06-08240-2.

[24]

S. Ibrahim and R. Jrad, Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain,, J. Differential Equations, 250 (2011), 3740. doi: 10.1016/j.jde.2011.01.008.

[25]

S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Well posedness and unconditional non uniqueness for a 2D semilinear heat equation,, preprint., ().

[26]

M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 297. doi: 10.1016/j.anihpc.2013.03.004.

[27]

T. Kato, Schrödinger operators with singular potentials,, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 13 (1972), 135. doi: 10.1007/BF02760233.

[28]

J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma,, Phys. Fluids, 20 (1977), 1176. doi: 10.1063/1.861679.

[29]

K. Morii, T. Sato and H. Wadade, Brézis-Gallouët-Wainger inequality with a double logarithmic term on a bounded domain and its sharp constants,, Math. Inequal. Appl., 14 (2011), 295. doi: 10.7153/mia-14-24.

[30]

K. Morii, T. Sato and H. Wadade, Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: its sharp constants and extremal functions,, Nonlinear Anal., 73 (2010), 1747. doi: 10.1016/j.na.2010.05.012.

[31]

K. Morii, T. Sato, Y. Sawano and H. Wadade, Sharp constants of Brézis-Gallouët-Wainger type inequalities with a double logarithmic term on bounded domains in Besov and Triebel-Lizorkin spaces,, Boundary Value Problems, (2010).

[32]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077.

[33]

S. Nagayasu and H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin,, J. Funct. Anal., 258 (2010), 3725. doi: 10.1016/j.jfa.2010.02.015.

[34]

M. Nakamura, Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces,, Reviews in Mathematical Physics, 23 (2011), 903. doi: 10.1142/S0129055X11004473.

[35]

M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, J. Funct. Anal., 155 (1998), 364. doi: 10.1006/jfan.1997.3236.

[36]

M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth,, Math. Z., 231 (1999), 479. doi: 10.1007/PL00004737.

[37]

M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces,, Publ. Res. Inst. Math. Sci., 37 (2001), 255. doi: 10.2977/prims/1145477225.

[38]

T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equation,, Nonlinear Anal., 14 (1990), 765. doi: 10.1016/0362-546X(90)90104-O.

[39]

T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem,, J. Math. Anal. Appl., 155 (1991), 531. doi: 10.1016/0022-247X(91)90017-T.

[40]

T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012.

[41]

T. Ozawa, Characterization of Trudinger's inequality,, J. Inequal. Appl., 1 (1997), 369. doi: 10.1155/S102558349700026X.

[42]

I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term,, Arch. Rational Mech. Anal., 129 (1995), 201. doi: 10.1007/BF00383673.

[43]

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth,, Internat. Math. Res. Notices, (1994). doi: 10.1155/S1073792894000346.

[44]

R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, Indiana Univ. Math. J., 21 (1972), 841.

[45]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. Henri Poincaré, 5 (1988), 425.

[46]

M. Struwe, The critical nonlinear wave equation in two space dimensions,, J. Eur. Math. Soc. (JEMS), 15 (2013), 1805. doi: 10.4171/JEMS/404.

[47]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.

[48]

B. Wang, Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$,, Discrete Contin. Dynam. Systems, 5 (1999), 753. doi: 10.3934/dcds.1999.5.753.

[49]

Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy,, Proc. Amer. Math. Soc., 136 (2008), 3477. doi: 10.1090/S0002-9939-08-09514-2.

show all references

References:
[1]

S. Adachi and K. Tanaka, A scale-invariant form of Trudinger-Moser inequality and its best exponent,, Proc. Amer. Math. Soc., 1102 (1999), 148.

[2]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.1090/S0894-0347-1989-1002633-4.

[3]

L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth,, Comm. Partial Differential Equations, 36 (2011), 797. doi: 10.1080/03605302.2010.534684.

[4]

A. A. Baraket, Local existence and estimations for a semilinear wave equation in two dimension space,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7 (2004), 1.

[5]

P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.1090/S0002-9947-1984-0742415-3.

[6]

C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988).

[7]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, (1976).

[8]

M. Bouchekif and A. Matallah, Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev-Hardy exponent,, Appl. Math. Lett., 22 (2009), 268. doi: 10.1016/j.aml.2008.03.024.

[9]

P. Brenner, $L_p$-estimates of difference schemes for strictly hyperbolic systems with nonsmooth data,, SIAM J. Numer. Anal., 14 (1977), 1126. doi: 10.1137/0714078.

[10]

H. Brézis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential,, Selecta Math. (N.S.), 11 (2005), 1. doi: 10.1007/s00029-005-0003-z.

[11]

H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: 10.1016/0362-546X(80)90068-1.

[12]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Partial Differential Equations, 5 (1980), 773. doi: 10.1080/03605308008820154.

[13]

N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential,, J. Funct. Anal., 203 (2003), 519. doi: 10.1016/S0022-1236(03)00238-6.

[14]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259.

[15]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 1267. doi: 10.3934/cpaa.2014.13.1267.

[16]

J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ., 6 (2009), 549. doi: 10.1142/S0219891609001927.

[17]

Z. Gan, Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential,, Commun. Pure Appl. Anal., 8 (2009), 1541. doi: 10.3934/cpaa.2009.8.1541.

[18]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 441. doi: 10.1006/jdeq.1997.3375.

[19]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation. II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15.

[20]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119.

[21]

T-S. Hsu and H-L. Lin, Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev-Hardy exponents,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 617. doi: 10.1017/S0308210509000729.

[22]

S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity,, Comm. Pure Appl. Math., 59 (2006), 1639. doi: 10.1002/cpa.20127.

[23]

S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant,, Proc. Amer. Math. Soc., 135 (2007), 87. doi: 10.1090/S0002-9939-06-08240-2.

[24]

S. Ibrahim and R. Jrad, Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain,, J. Differential Equations, 250 (2011), 3740. doi: 10.1016/j.jde.2011.01.008.

[25]

S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Well posedness and unconditional non uniqueness for a 2D semilinear heat equation,, preprint., ().

[26]

M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 297. doi: 10.1016/j.anihpc.2013.03.004.

[27]

T. Kato, Schrödinger operators with singular potentials,, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 13 (1972), 135. doi: 10.1007/BF02760233.

[28]

J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma,, Phys. Fluids, 20 (1977), 1176. doi: 10.1063/1.861679.

[29]

K. Morii, T. Sato and H. Wadade, Brézis-Gallouët-Wainger inequality with a double logarithmic term on a bounded domain and its sharp constants,, Math. Inequal. Appl., 14 (2011), 295. doi: 10.7153/mia-14-24.

[30]

K. Morii, T. Sato and H. Wadade, Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: its sharp constants and extremal functions,, Nonlinear Anal., 73 (2010), 1747. doi: 10.1016/j.na.2010.05.012.

[31]

K. Morii, T. Sato, Y. Sawano and H. Wadade, Sharp constants of Brézis-Gallouët-Wainger type inequalities with a double logarithmic term on bounded domains in Besov and Triebel-Lizorkin spaces,, Boundary Value Problems, (2010).

[32]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077.

[33]

S. Nagayasu and H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin,, J. Funct. Anal., 258 (2010), 3725. doi: 10.1016/j.jfa.2010.02.015.

[34]

M. Nakamura, Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces,, Reviews in Mathematical Physics, 23 (2011), 903. doi: 10.1142/S0129055X11004473.

[35]

M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, J. Funct. Anal., 155 (1998), 364. doi: 10.1006/jfan.1997.3236.

[36]

M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth,, Math. Z., 231 (1999), 479. doi: 10.1007/PL00004737.

[37]

M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces,, Publ. Res. Inst. Math. Sci., 37 (2001), 255. doi: 10.2977/prims/1145477225.

[38]

T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equation,, Nonlinear Anal., 14 (1990), 765. doi: 10.1016/0362-546X(90)90104-O.

[39]

T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem,, J. Math. Anal. Appl., 155 (1991), 531. doi: 10.1016/0022-247X(91)90017-T.

[40]

T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012.

[41]

T. Ozawa, Characterization of Trudinger's inequality,, J. Inequal. Appl., 1 (1997), 369. doi: 10.1155/S102558349700026X.

[42]

I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term,, Arch. Rational Mech. Anal., 129 (1995), 201. doi: 10.1007/BF00383673.

[43]

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth,, Internat. Math. Res. Notices, (1994). doi: 10.1155/S1073792894000346.

[44]

R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, Indiana Univ. Math. J., 21 (1972), 841.

[45]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. Henri Poincaré, 5 (1988), 425.

[46]

M. Struwe, The critical nonlinear wave equation in two space dimensions,, J. Eur. Math. Soc. (JEMS), 15 (2013), 1805. doi: 10.4171/JEMS/404.

[47]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.

[48]

B. Wang, Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$,, Discrete Contin. Dynam. Systems, 5 (1999), 753. doi: 10.3934/dcds.1999.5.753.

[49]

Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy,, Proc. Amer. Math. Soc., 136 (2008), 3477. doi: 10.1090/S0002-9939-08-09514-2.

[1]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

[2]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[3]

Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024

[4]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110

[5]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

[6]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212

[7]

B. Abdellaoui, I. Peral. On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2003, 2 (4) : 539-566. doi: 10.3934/cpaa.2003.2.539

[8]

Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629

[9]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002

[10]

Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963

[11]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697

[12]

Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378

[13]

Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741

[14]

Jijiang Sun, Shiwang Ma. Infinitely many sign-changing solutions for the Brézis-Nirenberg problem. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2317-2330. doi: 10.3934/cpaa.2014.13.2317

[15]

Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455

[16]

Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031

[17]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903

[18]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[19]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[20]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

[Back to Top]