Previous |  Up |  Next

Article

Keywords:
quasilinear elliptic equations; weak solutions; solvability
Summary:
We discuss how the choice of the functional setting and the definition of the weak solution affect the existence and uniqueness of the solution to the equation \[ -\Delta _p u = f \ \text{in} \ \Omega , \] where $\Omega $ is a very general domain in $\mathbb{R}^N$, including the case $\Omega = \mathbb{R}^N$.
References:
[1] P. Drábek: Solvability and Bifurcations of Nonlinear Equations. Pitman Research Notes in Mathematics Series 264, Longman, 1992. MR 1175397
[2] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities, de Gruyter Series in Nonlinear Analysis and Applications 5. Walter de Gruyter, Berlin, 1997. MR 1460729
[3] P. Drábek, C. G. Simader: Nonlinear eigenvalue problem for quasilinear equations in unbounded domains. Math. Nachrichten 203 (1999), 5–30. MR 1698637
[4] S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier, Amsterdam, 1980. MR 0558764
[5] S. Fučík, J. Nečas, J. Souček, V. Souček: Spectral Analysis of Nonlinear Operators. Lecture Notes in Mathematics 346, Springer, Berlin, 1973. MR 0467421
[6] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1977. MR 0473443
[7] V. Goldshtein, M. Troyanov: Sur la non résolubilité du $p$-laplacien C.R. Acad. Sci. Paris, t. 326, Sér. I (1998), 1185–1187. MR 1650266
[8] A. Kufner, O. John, S. Fučík: Function Spaces. Academia, Praha, 1977. MR 0482102
[9] J. Naumann, C. G.Simader: A second look on definition and equivalent norms of Sobolev spaces. Math. Bohem. 124 (1999), 315–328. MR 1780700 | Zbl 0941.46019
[10] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Praha, 1967. MR 0227584
[11] C. G. Simader: Sobolev’s original definition of his spaces revisited and a comparison with nowadays definition. Le Matematiche 54 (1999), 149–178. MR 1749828 | Zbl 0947.46022
[12] C. G. Simader, H. Sohr: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series 360, Addison Wesley Longman, 1996. MR 1454361
Partner of
EuDML logo