Existence and nonexistence of least energy nodal solution for a class of elliptic problem in R2

Claudianor Oliveira Alves, Denilson Pereira

DOI: http://dx.doi.org/10.12775/TMNA.2015.078


In this work, we prove the existence of least energy nodal solutions for a class of elliptic problem in both cases, bounded and unbounded domain, when the nonlinearity has exponential critical growth in $\mathbb{R}^2$. Moreover, we also prove a nonexistence result of least energy nodal solution for the autonomous case in whole $\mathbb{R}^{2}$.


Variational methods; exponential critical growth; nodal solution

Full Text:



Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of R2 involving critical exponent, Ann. Scuola. Norm. Sup. Pisa 17 (1990), 481-504.

C.O. Alves, Multiplicity of solutions for a class of elliptic problem in R2 with Neumann conditions, J. Differential Equations 219 (2005), 20-39.

C.O. Alves, P.C. Carri~ao and E. S. Medeiros, Multiplicity of solution for a class of quasilinear problem in exterior domain with Neumann conditions, Abstr. Appl. Anal. 3 (2004), 251-268.

C.O. Alves, J.M. do O and O.H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth, Nonlinear Anal. 56 (2004), 781-791.

C.O. Alves and S.H.M. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrodinger equations, J. Math. Anal. Appl. 296 (2004), 563-577.

C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations 234 (2007), 464-484.

T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer and 2-nodal solutions of a semiclassical nonlinear Schrodinger equation, Math. Ann. 338 (2007), 147-185

T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrodinger equations, Comm. Partial Differential Equations 29 (2004), 25-42.

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

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincare Anal. Non Lineaire. 22 (2005), 259-281.

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.

T. Barstch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on RN , Arch. Rat. Mech. Anal. 124 (1993), 261-276.

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN. Comm. Partial Differential Equations 20 (1995), 1725-1741.

, Sign changing solutions of nonlinear Schrodinger equations, Topol. Methods Nonlinear Anal. 13 (1999), 191-198.

H. Brezis, Analyse fonctionnelle thorie et applications, Masson, Paris, 1983.

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

D.M. Cao and E.S. Noussair, Multiplicity of positive and nodal solutions for a nonlinear elliptic problem in RN, Ann. Inst. H. Poincare, Section C, 5 (1996), 567-588.

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

G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), no. 3, 289-306.

D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. 3 (1995), 139-153.

C. Miranda, Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital. Ser. II, Anno III, n. 1 19 (1940), 5-7.

J. Moser, A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J. (20) (1971), 1077-1092.

E.S. Noussair and J. Wei, On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Ind. Univ. Math. J. 46 (1997), 1255-1271.

E.S. Noussair and J. Wei, On the location of spikes and profile of nodal solutions for a singularly perturbed Neumann problem, Comm. Partial Differential Equations 23 (1998), 793-816.

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

M. Willem, Minimax Theorems, Birkhauser, 1996.

W. Zou, Sign-Changing Critical Point Theory, Springer, Berlin, 2008.


  • There are currently no refbacks.

Partnerzy platformy czasopism