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

Federica Sani ^1,

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 and 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.
[2]	Adimurthi, Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in $\mathbb{R}^{2}$, Proc. Indian Acad. Sci. Math. Sci., 99 (1989), 49-73. doi: 10.1007/BF02874647.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. Chabrowski and J. M. Do Ó, On some fourth-order semilinear elliptic problems in $\mathbb{R}^{N}$, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.
[11]	D. G. Costa, On a class of elliptic systems in $R^N$, Electon. J. Differential Equations, 1994, No. 07, approx. 14 pp. (electronic).
[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.
[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.
[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.
[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.
[16]	E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.
[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.
[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.
[19]	R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[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.
[21]	P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83.
[22]	H. L. Royden, "Real Analysis," Third edition, Macmillan Publishing Company, New York, 1988.
[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.
[24]	B. Ruf and F. Sani, Sharp Adams-type inequalities in $R^n$, Trans. Amer. Math. Soc., in press, http://dx.doi.org/10.1090/S0002-9947-2012-05561-9
[25]	F. Sani, A biharmonic equation in $R^4$ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904.

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References:

[1]	D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Annals of Math., 128 (1988), 385-398. doi: 10.2307/1971445.
[2]	Adimurthi, Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in $\mathbb{R}^{2}$, Proc. Indian Acad. Sci. Math. Sci., 99 (1989), 49-73. doi: 10.1007/BF02874647.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. Chabrowski and J. M. Do Ó, On some fourth-order semilinear elliptic problems in $\mathbb{R}^{N}$, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.
[11]	D. G. Costa, On a class of elliptic systems in $R^N$, Electon. J. Differential Equations, 1994, No. 07, approx. 14 pp. (electronic).
[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.
[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.
[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.
[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.
[16]	E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.
[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.
[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.
[19]	R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[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.
[21]	P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83.
[22]	H. L. Royden, "Real Analysis," Third edition, Macmillan Publishing Company, New York, 1988.
[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.
[24]	B. Ruf and F. Sani, Sharp Adams-type inequalities in $R^n$, Trans. Amer. Math. Soc., in press, http://dx.doi.org/10.1090/S0002-9947-2012-05561-9
[25]	F. Sani, A biharmonic equation in $R^4$ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904.

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