# American Institute of Mathematical Sciences

June  2012, 32(6): 2187-2205. doi: 10.3934/dcds.2012.32.2187

## Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth

 1 Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

Received  April 2011 Revised  June 2011 Published  February 2012

The main purpose of this paper is to establish the existence of nontrivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adams inequality [1] of Moser-Trudinger type. More precisely, we consider the semilinear elliptic equation $\left( -\Delta\right) ^{m}u=f(x,u),$ subject to the Dirichlet boundary condition $u=\nabla u=...=\nabla^{m-1}u=0$, on the bounded domains $\Omega\subset \mathbb{R}^{2m}$ when the nonlinear term $f$ satisfies exponential growth condition. We will study the above problem both in the case when $f$ satisfies the well-known Ambrosetti-Rabinowitz condition and in the case without the Ambrosetti-Rabinowitz condition. This is one of a series of works by the authors on nonlinear equations of Laplacian in $\mathbb{R}^2$ and $N-$Laplacian in $\mathbb{R}^N$ when the nonlinear term has the exponential growth and with a possible lack of the Ambrosetti-Rabinowitz condition (see [23], [24]).
Citation: Nguyen Lam, Guozhen Lu. Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2187-2205. doi: 10.3934/dcds.2012.32.2187
##### References:
 [1] David R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2), 128 (1988), 385-398.  Google Scholar [2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17 (1990), 393-413.  Google Scholar [3] Antonio Ambrosetti and Paul 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 [4] Gianni Arioli, Filippo Gazzola, Hans-Christoph Grunau and Enzo Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal., 36 (2005), 1226-1258. doi: 10.1137/S0036141002418534.  Google Scholar [5] Elvise Berchio, Filippo Gazzola and Enzo Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23. doi: 10.1016/j.jde.2006.04.003.  Google Scholar [6] Elvise Berchio, Filippo Gazzola and Tobias Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations, 12 (2007), 381-406.  Google Scholar [7] Jiguang Bao, Nguyen Lam and Guozhen Lu, Existence and regularity of solutions to polyharmonic equations with critical exponential growth in the whole space,, to appear., ().   Google Scholar [8] Haim Brézis and Louis 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 [9] Lennart Carleson and Sun-Yung A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2), 110 (1986), 113-127.  Google Scholar [10] Sun-Yung A. Chang and Paul C. Yang, The inequality of Moser and Trudinger and applications to conformal geometry, Dedicated to the memory of Jurgen K. Moser, Comm. Pure Appl. Math., 56 (2003), 1135-1150. doi: 10.1002/cpa.3029.  Google Scholar [11] Giovanna Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian), Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336.  Google Scholar [12] Giovanna Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, (Italian), Ann. Mat. Pura Appl. (4), 124 (1980), 161-179. doi: 10.1007/BF01795391.  Google Scholar [13] J. M. B. do Ó, Semilinear Dirichlet problems for the N-Laplacian in $\mathbbR^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.  Google Scholar [14] 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 [15] Martin Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.  Google Scholar [16] Luigi Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828.  Google Scholar [17] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbbR^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  Google Scholar [18] Filippo Gazzola, Hans-Christoph Grunau and Marco Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  Google Scholar [19] Filippo Gazzola, Hans-Christoph Grunau and Guido Sweers, "Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains," Lecture Notes in Mathematics, 1991, Springer-Verlag, Berlin, 2010.  Google Scholar [20] Hans-Christoph Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 243-252.  Google Scholar [21] Hans-Christoph Grunau and Guido Sweers, Classical solutions for some higher order semilinear elliptic equations under weak growth conditions, Nonlinear Anal., 28 (1997), 799-807. doi: 10.1016/0362-546X(95)00194-Z.  Google Scholar [22] Omar Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth, Adv. Differential Equations, 4 (1999), 877-906.  Google Scholar [23] Nguyen Lam and Guozhen Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition,, to appear., ().   Google Scholar [24] Nguyen Lam and Guozhen Lu, $N-$Laplacian equations in $\mathbbR^N$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition,, \arXiv{1012.5489}., ().   Google Scholar [25] Nguyen Lam and Guozhen Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbbR^N$, Journal of Functional Analysis, 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012.  Google Scholar [26] M. Lazzo and P. G. Schmidt, Oscillatory radial solutions for subcritical biharmonic equations, J. Differential Equations, 247 (2009), 1479-1504. doi: 10.1016/j.jde.2009.05.005.  Google Scholar [27] Mark Leckband, Moser's inequality on the ball $B^n$ for functions with mean value zero, Comm. Pure Appl. Math., 58 (2005), 789-798. doi: 10.1002/cpa.20056.  Google Scholar [28] Yuxiang Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192.  Google Scholar [29] Yuxiang Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050.  Google Scholar [30] Yuxiang Li and Cheikh B. Ndiaye, Extremal functions for Moser-Trudinger type inequality on compact closed $4$-manifolds, J. Geom. Anal., 17 (2007), 669-699. doi: 10.1007/BF02937433.  Google Scholar [31] Yuxiang Li and Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbbR^n$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137.  Google Scholar [32] Kai-Chin Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3.  Google Scholar [33] Guozhen Lu and Yunyan Yang, A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in $\mathbbR^2$, Nonlinear Anal., 70 (2009), 2992-3001. doi: 10.1016/j.na.2008.12.022.  Google Scholar [34] Guozhen Lu and Yunyan 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 [35] Guozhen Lu and Yunyan Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving Lp norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  Google Scholar [36] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.  Google Scholar [37] Jurgen Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar [38] S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar [39] Patrizia Pucci and James Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.  Google Scholar [40] Wolfgang Reichel and Tobias Weth, Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems, J. Differential Equations, 248 (2010), 1866-1878. doi: 10.1016/j.jde.2009.09.012.  Google Scholar [41] Bernard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbbR^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar [42] Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  Google Scholar [43] Yunyan Yang, A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776. doi: 10.1090/S0002-9947-07-04272-9.  Google Scholar

show all references

##### References:
 [1] David R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2), 128 (1988), 385-398.  Google Scholar [2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17 (1990), 393-413.  Google Scholar [3] Antonio Ambrosetti and Paul 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 [4] Gianni Arioli, Filippo Gazzola, Hans-Christoph Grunau and Enzo Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal., 36 (2005), 1226-1258. doi: 10.1137/S0036141002418534.  Google Scholar [5] Elvise Berchio, Filippo Gazzola and Enzo Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23. doi: 10.1016/j.jde.2006.04.003.  Google Scholar [6] Elvise Berchio, Filippo Gazzola and Tobias Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations, 12 (2007), 381-406.  Google Scholar [7] Jiguang Bao, Nguyen Lam and Guozhen Lu, Existence and regularity of solutions to polyharmonic equations with critical exponential growth in the whole space,, to appear., ().   Google Scholar [8] Haim Brézis and Louis 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 [9] Lennart Carleson and Sun-Yung A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2), 110 (1986), 113-127.  Google Scholar [10] Sun-Yung A. Chang and Paul C. Yang, The inequality of Moser and Trudinger and applications to conformal geometry, Dedicated to the memory of Jurgen K. Moser, Comm. Pure Appl. Math., 56 (2003), 1135-1150. doi: 10.1002/cpa.3029.  Google Scholar [11] Giovanna Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian), Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336.  Google Scholar [12] Giovanna Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, (Italian), Ann. Mat. Pura Appl. (4), 124 (1980), 161-179. doi: 10.1007/BF01795391.  Google Scholar [13] J. M. B. do Ó, Semilinear Dirichlet problems for the N-Laplacian in $\mathbbR^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.  Google Scholar [14] 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 [15] Martin Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.  Google Scholar [16] Luigi Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828.  Google Scholar [17] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbbR^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  Google Scholar [18] Filippo Gazzola, Hans-Christoph Grunau and Marco Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  Google Scholar [19] Filippo Gazzola, Hans-Christoph Grunau and Guido Sweers, "Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains," Lecture Notes in Mathematics, 1991, Springer-Verlag, Berlin, 2010.  Google Scholar [20] Hans-Christoph Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 243-252.  Google Scholar [21] Hans-Christoph Grunau and Guido Sweers, Classical solutions for some higher order semilinear elliptic equations under weak growth conditions, Nonlinear Anal., 28 (1997), 799-807. doi: 10.1016/0362-546X(95)00194-Z.  Google Scholar [22] Omar Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth, Adv. Differential Equations, 4 (1999), 877-906.  Google Scholar [23] Nguyen Lam and Guozhen Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition,, to appear., ().   Google Scholar [24] Nguyen Lam and Guozhen Lu, $N-$Laplacian equations in $\mathbbR^N$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition,, \arXiv{1012.5489}., ().   Google Scholar [25] Nguyen Lam and Guozhen Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbbR^N$, Journal of Functional Analysis, 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012.  Google Scholar [26] M. Lazzo and P. G. Schmidt, Oscillatory radial solutions for subcritical biharmonic equations, J. Differential Equations, 247 (2009), 1479-1504. doi: 10.1016/j.jde.2009.05.005.  Google Scholar [27] Mark Leckband, Moser's inequality on the ball $B^n$ for functions with mean value zero, Comm. Pure Appl. Math., 58 (2005), 789-798. doi: 10.1002/cpa.20056.  Google Scholar [28] Yuxiang Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192.  Google Scholar [29] Yuxiang Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050.  Google Scholar [30] Yuxiang Li and Cheikh B. Ndiaye, Extremal functions for Moser-Trudinger type inequality on compact closed $4$-manifolds, J. Geom. Anal., 17 (2007), 669-699. doi: 10.1007/BF02937433.  Google Scholar [31] Yuxiang Li and Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbbR^n$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137.  Google Scholar [32] Kai-Chin Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3.  Google Scholar [33] Guozhen Lu and Yunyan Yang, A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in $\mathbbR^2$, Nonlinear Anal., 70 (2009), 2992-3001. doi: 10.1016/j.na.2008.12.022.  Google Scholar [34] Guozhen Lu and Yunyan 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 [35] Guozhen Lu and Yunyan Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving Lp norm in two dimension, Discrete Contin. Dyn. Syst., 25 (2009), 963-979.  Google Scholar [36] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.  Google Scholar [37] Jurgen Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.  doi: 10.1512/iumj.1971.20.20101.  Google Scholar [38] S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar [39] Patrizia Pucci and James Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.  Google Scholar [40] Wolfgang Reichel and Tobias Weth, Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems, J. Differential Equations, 248 (2010), 1866-1878. doi: 10.1016/j.jde.2009.09.012.  Google Scholar [41] Bernard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbbR^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar [42] Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  Google Scholar [43] Yunyan Yang, A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776. doi: 10.1090/S0002-9947-07-04272-9.  Google Scholar
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