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

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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

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

Received May 2011 Revised December 2011 Published September 2012

Abstract
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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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. doi: 10.1007/BF00381236. Google Scholar
[16]	E. H. Lieb and M. Loss, "Analysis,", Second edition, (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. 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. doi: 10.1512/iumj.1971.20.20101. Google Scholar
[19]	R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. doi: 10.1007/BF01941322. Google Scholar
[20]	P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. 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. Google Scholar
[22]	H. L. Royden, "Real Analysis,", Third edition, (1988). Google Scholar
[23]	B. Ruf, A sharp trudinger-moser type inequality for unbounded domains in $R^2$,, J. Funct. Anal., 219 (2005), 340. 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. Google Scholar

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References:

[1]	D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,, Annals of Math., 128 (1988), 385. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. doi: 10.1007/BF00381236. Google Scholar
[16]	E. H. Lieb and M. Loss, "Analysis,", Second edition, (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. 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. doi: 10.1512/iumj.1971.20.20101. Google Scholar
[19]	R. S. Palais, The principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. doi: 10.1007/BF01941322. Google Scholar
[20]	P. Pucci and J. Serrin, A general variational identity,, Indiana Univ. Math. J., 35 (1986), 681. 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. Google Scholar
[22]	H. L. Royden, "Real Analysis,", Third edition, (1988). Google Scholar
[23]	B. Ruf, A sharp trudinger-moser type inequality for unbounded domains in $R^2$,, J. Funct. Anal., 219 (2005), 340. 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. Google Scholar

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