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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

J. M. B. do Ó, Semilinear Dirichlet problems for the N-Laplacian in $\mathbb{R}^N2$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.

[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.

[15]

Martin Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.

[16]

Luigi Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828.

[17]

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

[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.

[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.

[20]

Hans-Christoph Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 243-252.

[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.

[22]

Omar Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth, Adv. Differential Equations, 4 (1999), 877-906.

[23]

Nguyen Lam and Guozhen Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, to appear.

[24]

Nguyen Lam and Guozhen Lu, $N-$Laplacian equations in $\mathbb{R}^N2$ with subcritical and critical growth without the Ambrosetti-Rabinowitz conditionarXiv:1012.5489.

[25]

Nguyen Lam and Guozhen Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbb{R}^N2$, Journal of Functional Analysis, 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012.

[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.

[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.

[28]

Yuxiang Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192.

[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.

[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.

[31]

Yuxiang Li and Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137.

[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.

[33]

Guozhen Lu and Yunyan Yang, A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in $\mathbb{R}^2$, Nonlinear Anal., 70 (2009), 2992-3001. doi: 10.1016/j.na.2008.12.022.

[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.

[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.

[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.

[37]

Jurgen Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101.

[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.

[39]

Patrizia Pucci and James Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.

[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.

[41]

Bernard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.

[42]

Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

J. M. B. do Ó, Semilinear Dirichlet problems for the N-Laplacian in $\mathbb{R}^N2$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.

[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.

[15]

Martin Flucher, Extremal functions for the Trudinger-Moser inequality in $2$ dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514.

[16]

Luigi Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., 68 (1993), 415-454. doi: 10.1007/BF02565828.

[17]

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

[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.

[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.

[20]

Hans-Christoph Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 243-252.

[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.

[22]

Omar Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth, Adv. Differential Equations, 4 (1999), 877-906.

[23]

Nguyen Lam and Guozhen Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, to appear.

[24]

Nguyen Lam and Guozhen Lu, $N-$Laplacian equations in $\mathbb{R}^N2$ with subcritical and critical growth without the Ambrosetti-Rabinowitz conditionarXiv:1012.5489.

[25]

Nguyen Lam and Guozhen Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbb{R}^N2$, Journal of Functional Analysis, 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012.

[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.

[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.

[28]

Yuxiang Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192.

[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.

[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.

[31]

Yuxiang Li and Bernhard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Indiana Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137.

[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.

[33]

Guozhen Lu and Yunyan Yang, A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in $\mathbb{R}^2$, Nonlinear Anal., 70 (2009), 2992-3001. doi: 10.1016/j.na.2008.12.022.

[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.

[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.

[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.

[37]

Jurgen Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. doi: 10.1512/iumj.1971.20.20101.

[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.

[39]

Patrizia Pucci and James Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.

[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.

[41]

Bernard Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.

[42]

Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.

[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.

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