Nodal solution for a planar problem with fast increasing weights

Giovany M. Figueiredo, Marcelo F. Furtado, Ricardo Ruviaro



In this paper we prove the existence of a sign-changing solutions for the equation $$ -\Delta u - \frac{1}{2} ( x \cdot \nabla u) = f(u), \quad x \in \mathbb{R}^2, $$ where $f$ has exponential critical growth in the sense of the Trudinger-Moser inequality. In the proof we apply variational methods.


Nodal solutions; critical exponential growth; self-similar solutions

Full Text:



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.

F.V. Atkinson and L.A. Peletier, Sur les solutions radiales de l’équation ∆u + x · ∇u/2 + λu/2 + |u|p−1 u = 0, (in French); On the radial solutions of the equation ∆u + x ∇u/2 + λu/2 + |u|p−1 u = 0, C.R. Acad. Sci. Paris Sér. I Math. 302 (1986), 99–101.

T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods in Nonlinear Anal. 22 (2003), 1–14.

T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1–18.

D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2 , Comm. Partial Differential Equations 17 (1992), 407–435.

A. Castro, J. Cossio and J. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041–1053.

F. Catrina, M. Furtado and M. Montenegro, Positive solutions for nonlinear elliptic equations with fast increasing weights, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1157–1178.

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 (1987), 1103–1133.

L.C. Ferreira, M.F. Furtado and E.S. Medeiros, Existence and multiplicity of selfsimilar solutions for heat equations with nonlinear boundary conditions, Calc. Var. Partial Differential Equations 54 (2015), 4065–4078.

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 4 (1995), 139–153.

D.G. de Figueiredo, J.M.B. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2002), 135–152.

M.F. Furtado, E.S. Medeiros and U. Severo, On a class of semilinear elliptic eigenvalue problems in R2 , Proc. Edinburgh Math. Soc. 60 (2017), 107–126.

M.F. Furtado, E.S. Medeiros and U.B. Severo, A Trudinger–Moser inequality in a weighted Sobolev space and applications, Math. Nachr. 287 (2014), 1255–1273.

M.F. Furtado, O.H. Miyagaki and J.P. Silva, On a class of nonlinear elliptic equations with fast increasing weight and critical growth, J. Differential Equations 249 (2010), 1035–1055.

A. Haraux and F.B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167–189.

L. Herraiz, Asymptotic behavior of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 49–105.

N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation, Math. Ann. 307 (1997), 663–675.

C. Miranda, Un’ osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital. 3 (1940), 5–7.

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1985), 185–201.

Y. Naito, Self-similar solutions for a semilinear heat equation with critical Sobolev exponent, Indiana Univ. Math. J. 57 (2008), 1283–1315.

Y. Naito and T. Suzuki, Radial symmetry of self-similar solutions for semilinear heat equations, J. Differential Equations 163 (2000), 407–428.

J.M.B. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.

B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in R2 , J. Funct. Anal. 219 (2005), 340–367.

N.S. Trudinger, On the imbedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.

M. Willem, Minimax Theorems, Birkhäuser, Boston, Basel, Berlim, (1996).

Q. Xiaotao and C. Jianqing, Sign-changing solutions for elliptic equations with fast increasing weight and concave-convex nonlinearities, Electron. J. Differential Equations (2017), paper no. 229, 16 pp.


  • There are currently no refbacks.

Partnerzy platformy czasopism