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 \(\alpha\).


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
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[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
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