American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$
  • CPAA Home
  • This Issue
  • Next Article
    Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction
January  2013, 12(1): 405-428. doi: 10.3934/cpaa.2013.12.405

A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth

Federica Sani 1,

1. 

Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano, Italy

Received  May 2011 Revised  December 2011 Published  September 2012

In this paper we give sufficient conditions for the existence of solutions of a biharmonic equation of the form

$ \Delta^2 u + V(x)u = f(u)$ in $\mathbb{R}^4$

where $V$ is a continuous positive potential bounded away from zero and the nonlinearity $f(s)$ behaves like $e^{\alpha_0 s^2}$ at infinity for some $\alpha_0>0$.
In order to overcome the lack of compactness due to the unboundedness of the domain $\mathbb{R}^4$, we require some additional assumptions on $V$. In the case when the potential $V$ is large at infinity we obtain the existence of a nontrivial solution, while requiring the potential $V$ to be spherically symmetric we obtain the existence of a nontrivial radial solution. In both cases, the main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$.

Keywords: Biharmonic equation, critical exponential growth, Trudinger-Moser inequalities, variational methods..
Mathematics Subject Classification: Primary: 35J3.
Citation: Federica Sani. A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 405-428. doi: 10.3934/cpaa.2013.12.405
References:
[1]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Annals of Math., 128 (1988), 385-398. doi: 10.2307/1971445.  Google Scholar

[2]

Adimurthi, Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in $\er^2$, Proc. Indian Acad. Sci. Math. Sci., 99 (1989), 49-73. doi: 10.1007/BF02874647.  Google Scholar

[3]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 393-413.  Google Scholar

[4]

C. O. Alves and J. M. Do Ó, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458.  Google Scholar

[5]

C. O. Alves, J. M. Do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal. Ser. A: Theory Methods, 46 (2001), 121-133. doi: 10.1016/S0362-546X(99)00449-6.  Google Scholar

[6]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[7]

T. Bartsch, M. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math., 571 (2004), 131-143. doi: 10.1515/crll.2004.037.  Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[9]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[10]

J. Chabrowski and J. M. Do Ó, On some fourth-order semilinear elliptic problems in $\er^N$, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[11]

D. G. Costa, On a class of elliptic systems in $R^N$, Electon. J. Differential Equations, 1994, No. 07, approx. 14 pp. (electronic).  Google Scholar

[12]

D. G. De Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  Google Scholar

[13]

J. M. Do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304. doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar

[14]

D. E. Edmunds, D. Fortunato and E. Jannelli, Fourth-order nonlinear elliptic equations with critical growth, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 83 (1989), 115-119.  Google Scholar

[15]

D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289. doi: 10.1007/BF00381236.  Google Scholar

[16]

E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[17]

G. Lu and Y. Yang, Adams' inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170. doi: 10.1016/j.aim.2008.10.011.  Google Scholar

[18]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Math. J., 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[19]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[20]

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83.  Google Scholar

[22]

H. L. Royden, "Real Analysis," Third edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[23]

B. Ruf, A sharp trudinger-moser type inequality for unbounded domains in $R^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[24]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $R^n$,, Trans. Amer. Math. Soc., (): 0002.   Google Scholar

[25]

F. Sani, A biharmonic equation in $R^4$ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904.  Google Scholar

show all references

References:
[1]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Annals of Math., 128 (1988), 385-398. doi: 10.2307/1971445.  Google Scholar

[2]

Adimurthi, Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in $\er^2$, Proc. Indian Acad. Sci. Math. Sci., 99 (1989), 49-73. doi: 10.1007/BF02874647.  Google Scholar

[3]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 393-413.  Google Scholar

[4]

C. O. Alves and J. M. Do Ó, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458.  Google Scholar

[5]

C. O. Alves, J. M. Do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal. Ser. A: Theory Methods, 46 (2001), 121-133. doi: 10.1016/S0362-546X(99)00449-6.  Google Scholar

[6]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[7]

T. Bartsch, M. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math., 571 (2004), 131-143. doi: 10.1515/crll.2004.037.  Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[9]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[10]

J. Chabrowski and J. M. Do Ó, On some fourth-order semilinear elliptic problems in $\er^N$, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.  Google Scholar

[11]

D. G. Costa, On a class of elliptic systems in $R^N$, Electon. J. Differential Equations, 1994, No. 07, approx. 14 pp. (electronic).  Google Scholar

[12]

D. G. De Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  Google Scholar

[13]

J. M. Do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304. doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar

[14]

D. E. Edmunds, D. Fortunato and E. Jannelli, Fourth-order nonlinear elliptic equations with critical growth, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 83 (1989), 115-119.  Google Scholar

[15]

D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289. doi: 10.1007/BF00381236.  Google Scholar

[16]

E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[17]

G. Lu and Y. Yang, Adams' inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math., 220 (2009), 1135-1170. doi: 10.1016/j.aim.2008.10.011.  Google Scholar

[18]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Math. J., 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101.  Google Scholar

[19]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[20]

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83.  Google Scholar

[22]

H. L. Royden, "Real Analysis," Third edition, Macmillan Publishing Company, New York, 1988.  Google Scholar

[23]

B. Ruf, A sharp trudinger-moser type inequality for unbounded domains in $R^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[24]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $R^n$,, Trans. Amer. Math. Soc., (): 0002.   Google Scholar

[25]

F. Sani, A biharmonic equation in $R^4$ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904.  Google Scholar

[1]

Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
[2]

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
[3]

Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455
[4]

Mengjie Zhang. Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1721-1735. doi: 10.3934/cpaa.2021038
[5]

Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 781-813. doi: 10.3934/dcds.2021137
[6]

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
[7]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452
[8]

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
[9]

Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047
[10]

Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191
[11]

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
[12]

Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure & Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
[13]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[14]

Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 809-823. doi: 10.3934/dcdss.2011.4.809
[15]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185
[16]

Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277
[17]

Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155
[18]

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
[19]

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
[20]

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

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy