On a class of quasilinear elliptic problems with critical exponential growth on the whole space
Keywords
Trudinger-Moser inequality, quasilinear elliptic equation, critical growth, radial operatorsAbstract
In this paper we prove a kind of weighted Trudinger-Moser inequality which is employed to establish sufficient conditions for the existence of solutions to a large class of quasilinear elliptic differential equations with critical exponential growth. The class of operators considered includes, as particular cases, the Laplace, $p$-Laplace and $k$-Hessian operators when acting on radially symmetric functions.References
C.O. Alves and G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , J. Differential Equations 246 (2009), 1288–1311.
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, Funct. Anal. 14 (1973), 349–381.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Rational Mech. Anal. 82 (1983), 313–346.
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
P. Clément, D.G. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal. 7 (1996), 133–170.
D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2, Comm. Partial Differential Equations 17 (1992), 407–435.
L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of pLaplace equations via the moving plane method, Arch. Rational Mech. Anal. 148 (1999), 291–308.
L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations 206 (2004), 483–515.
D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139–153.
D.G. de Figueiredo, J.V. Gonçalves and O.H. Miyagaki, On a class of quasilinear elliptic problems involving critical exponents, Commun. Contemp. Math. 2 (2000), 47–59.
Ph. Delanoë, Radially symmetric boundary value problems for real and complex elliptic Monge–Ampère equations, J. Differential Equations 58 (1985), 318–344.
J.F. de Oliveira and J.M. do Ó, Trudinger–Moser type inequalities for weighted Sobolev spaces involving fractional dimensions, Proc. Amer. Math. Soc. 142 (2014), 2813–2828.
J.F. de Oliveira and J.M. do Ó, Concentration-compactness principle and extremal functions for a sharp Trudinger–Moser inequality, Calc. Var. Partial Differential Equations 52 (2015), 125–163.
M. de Souza, On a singular elliptic problem involving critical growth in Rn , Nonlinear Differerential Equations Appl. 18 (2011), 199–215.
J.M. do Ó, Semilinear Dirichlet problems for the N -Laplacian in RN with nonlinearities in the critical growth range, Differential and Integral Equations (1996), 967–979.
J.M. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.
J.M. do Ó and J.F. de Oliveira, Concentration-compactness and extremal problems for a weighted Trudinger–Moser inequality, Commun. Contemp. Math., DOI: 10.1142/S0219199716500036.
J.M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in RN , J. Differential Equations 246 (2009), 1363–1386.
J. Dolbeault, P. Felmer and R. Monneau, Symmetry and non-uniformly elliptic operators, Differential Integral Equations 18 (2005), 141–154.
B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243.
B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn , Mathematical Analysis and Applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud. 7a, Academic Press, New York, 1981.
J. Jacobsen, A Liouville–Gelfand equation for k-Hessian operators, Rocky Mountain J. Math. 34 (2004), 665–683.
J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, Handbook of Differential Equations, pp. 359–435, Elsevier/North-Holland, Amsterdam, 2004.
J. Jacobsen and K. Schmitt, The Liouville–Bratu–Gelfand problem for radial operators, J. Differential Equations 184 (2002), 283–298.
N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N -Laplacian type with critical exponential growth in RN , J. Funct. Anal. 262 (2012), 1132–1165.
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092.
J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971), 304–318.
N.S. Trudinger and X.-J. Wang, Hessian measures I, Topol. Methods Nonlinear Anal. 10 (1997), 225–239.
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