The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the $N$-Laplacian $$ - \Delta_N u \equiv - \operatorname{div} (|\nabla u|^{N-2} \nabla u) = e(x,u) + h(x) \text{ in } \Omega $$ where $u \in W_0^{1,N}(\Bbb R^{N})$, $\Omega$ is a bounded smooth domain in $\Bbb R^{N}$, $N \geq 2$, $e(x,u)$ is a critical nonlinearity in the sense of the Trudinger-Moser inequality and $h(x) \in (W_0^{1,N})^*$ is a small perturbation.
@article{119123,
author = {Elliot Tonkes},
title = {Solutions to a perturbed critical semilinear equation concerning the $N$-Laplacian in $\Bbb R^{N}$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {40},
year = {1999},
pages = {679-699},
zbl = {1064.35511},
mrnumber = {1756545},
language = {en},
url = {http://dml.mathdoc.fr/item/119123}
}
Tonkes, Elliot. Solutions to a perturbed critical semilinear equation concerning the $N$-Laplacian in $\Bbb R^{N}$. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 679-699. http://gdmltest.u-ga.fr/item/119123/
Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Ann. Sc. Norm. Sup. Pisa, Series 4 17 (1990), 393-413. (1990) | MR 1079983 | Zbl 0732.35028
Some remarks on the Dirichlet problem with critical growth for the $n$-Laplacian, Houston J. Math. 17 (2) (1991), 285-298. (1991) | MR 1115150 | Zbl 0768.35015
Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. (1973) | MR 0370183 | Zbl 0273.49063
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (3) (1983), 486-490. (1983) | MR 0699419 | Zbl 0526.46037
On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), 113-127. (1986) | MR 0878016
On multiple solutions for the nonhomogeneous $p$-Laplacian with a critical Sobolev exponent, Differential Integral Equations 8 (4) (1995), 705-716. (1995) | MR 1306587 | Zbl 0814.35033
Existence and bifurcation of the positive solutions of a semilinear equation with critical exponent, J. Differential Equations 130 (1996), 179-200. (1996) | MR 1409029
Elliptic equations in $\Bbb R^{2}$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (2) (1995), 139-153. (1995) | MR 1386960
On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. (1974) | MR 0346619 | Zbl 0286.49015
Extremal functions for Moser's inequality, Trans. Amer. Math. Soc. 348 (7) (1996), 2663-2671. (1996) | MR 1333394
The Concentration Compactness Principle in the Calculus of Variations, part I, Rev. Mat. Iberoamericana 1 (1985), 185-201. (1985) | MR 0834360
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (11) (1971), 1077-1092. (1971) | MR 0301504
Semilinear Dirichlet problems for the $N$-Laplacian in ${\Bbb R}^N$ with nonlinearities in the critical growth range, Differential Integral Equations 9 (5) (1996), 967-979. (1996) | MR 1392090
Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, No. 65, AMS, 1986. | MR 0845785 | Zbl 0609.58002
On semilinear Neumann problems with critical growth for the $n$-Laplacian, Nonlinear Anal. 26 (1996), 1347-1366. (1996) | MR 1377667 | Zbl 0854.35045
On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (3) (1992), 281-304. (1992) | MR 1168304 | Zbl 0785.35046
On imbeddings into Orlicz spaces and some applications, Journal of Mathematics and Mechanics 17 (5) (1967), 473-483. (1967) | MR 0216286 | Zbl 0163.36402