×

Existence of critical points for some noncoercive functionals. (English) Zbl 1035.49007

Summary: We study critical points problems for some integral functionals with principal part having degenerate coerciveness, whose model is \[ J(v)=\frac {1}{2} \int_\Omega \frac {|\nabla v|^2} {\bigl(b(x)+ | v|\bigr)^{2\alpha}}-\frac {1}{m} \int_\Omega| v|^m,\quad v\in H^1_0(\Omega), \] with \(1<m<2^* (1-\alpha)\), \(2^*=2N/(N-2)\), \(\Omega\subset \mathbb{R}^N\) \((N>2)\) a bounded domain. We prove several existence and nonexistence results depending on different assumptions on both \(m\) and \(\alpha\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J10 Existence theories for free problems in two or more independent variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[2] Arcoya, D.; Boccardo, L., Critical points for multiple integrals of calculus of variations, Arch. Rat. Mech. Anal., 134, 249-274 (1996) · Zbl 0884.58023
[3] Arcoya, D.; Boccardo, L., Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl., 6, 79-100 (1999) · Zbl 0923.35049
[4] Arcoya D., Gamez J.L., Orsina L., Peral I., Local existence results for sub-super-critical elliptic problems. Comm. Appl. Anal., to appear; Arcoya D., Gamez J.L., Orsina L., Peral I., Local existence results for sub-super-critical elliptic problems. Comm. Appl. Anal., to appear · Zbl 1085.35511
[5] Boccardo, L.; Orsina, L., Existence and regularity of minima for integral functionals noncoercive in the energy space, Ann. Scuola Norm. Sup. Pisa, 25, 95-130 (1997) · Zbl 1015.49014
[6] Degiovanni, M.; Marzocchi, M., A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl., 167, 73-100 (1994) · Zbl 0828.58006
[7] De Giorgi, E., Teoremi di Semicontinuità nel Calcolo Delle Variazioni. Teoremi di Semicontinuità nel Calcolo Delle Variazioni, Lecture Notes (1968), Istituto Nazionale di Alta Matematica: Istituto Nazionale di Alta Matematica Roma
[8] Donato, P.; Giachetti, D., Quasilinear elliptic equations with quadratic growth in unbounded domains, Nonlinear Anal., 10, 791-804 (1986) · Zbl 0602.35036
[9] Ladyzenskaya, O. A.; Uralceva, N. N., Equations aux Dérivées Partielles de Type Elliptique (1968), Dunod: Dunod Paris · Zbl 0164.13001
[10] Pohožaev, S. I., Eigenfunctions of \(Δu + λf (u)=0\), Soviet Math. Dokl., 6, 1408-1411 (1965) · Zbl 0141.30202
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.