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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

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 $f$.

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

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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.  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
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