On a class of critical \(N\)-Laplacian problems. (English) Zbl 1497.35277
Summary: We establish some existence results for a class of critical \(N\)-Laplacian problems in a bounded domain in \(\mathbb{R}^N\). In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the \(\mathbb{Z}_2\)-cohomological index to obtain a non-trivial critical point.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35B33 | Critical exponents in context of PDEs |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
References:
| [1] | Adimurthi, A., 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(3) (1990), 393-413. · Zbl 0732.35028 |
| [2] | Adimurthi, A. and Yadava, S. L., Bifurcation results for semilinear elliptic problems with critical exponent in \(R^2\), Nonlinear Anal. 14(7) (1990), 607-612. · Zbl 0702.35015 |
| [3] | De Figueiredo, D. G., Miyagaki, O. H. and Ruf, B., Elliptic equations in \(R^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ. 3(2) (1995), 139-153. · Zbl 0820.35060 |
| [4] | De Figueiredo, D. G., Miyagaki, O. H. and Ruf, B., Corrigendum: Elliptic equations in \(R^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ. 4(2) (1996), 203. · Zbl 0847.35048 |
| [5] | De Figueiredo, D. G., Do Ó, J. M. and Ruf, B., On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55(2) (2002), 135-152. · Zbl 1040.35029 |
| [6] | De Figueiredo, D. G., Do Ó, J. M. and Ruf, B., Elliptic equations and systems with critical Trudinger-Moser nonlinearities, Discrete Contin. Dyn. Syst. 30(2) (2011), 455-476. · Zbl 1222.35084 |
| [7] | Degiovanni, M. and Lancelotti, S., Linking solutions for \(p\)-Laplace equations with nonlinearity at critical growth, J. Funct. Anal. 256(11) (2009), 3643-3659. · Zbl 1170.35418 |
| [8] | Do Ó, J. M. B., Semilinear Dirichlet problems for the \(N\)-Laplacian in \(\mathbb{R}^N\) with nonlinearities in the critical growth range, Differ. Int. Equ. 9(5) (1996), 967-979. · Zbl 0858.35043 |
| [9] | Fadell, E. R. and Rabinowitz, P. H., Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45(2) (1978), 139-174. · Zbl 0403.57001 |
| [10] | Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970), 1077/71-1092. · Zbl 0203.43701 |
| [11] | Perera, K., Nontrivial critical groups in \(p\)-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21(2) (2003), 301-309. · Zbl 1039.47041 |
| [12] | Perera, K., Agarwal, R. P. and O’Regan, D., Morse theoretic aspects of \(p\)-Laplacian type operators, Mathematical Surveys and Monographs, Volume 161 (American Mathematical Society, Providence, RI, 2010). · Zbl 1192.58007 |
| [13] | Rabinowitz, P. H., Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)5(1) (1978), 215-223. · Zbl 0375.35026 |
| [14] | Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. · Zbl 0163.36402 |
| [15] | Yang, Y. and Perera, K., \(N\)-Laplacian problems with critical Trudinger-Moser nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)16(4) (2016), 1123-1138. · Zbl 1359.35083 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
