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Concentration of low energy extremals. (English) Zbl 0938.35042

The authors investigate the asymptotic behaviour of solutions and approximate solutions \(u_\varepsilon\) of the variational problem \[ \sup\Big\{\int_\Omega F(u) : \int_\Omega |\nabla u|^2 \leq \varepsilon^2, \;\;u=0 \;\;on \;\;\partial\Omega\Big\} \] in the limit \(\varepsilon \rightarrow 0\). The integrand \(F\) (that can be nonconvex and discontinuous) is supposed to satisfy \(0\leq F(t)\leq c|t|^{2n/(n-2)}\) on a domain of dimension \(n\geq 3\). For smooth \(F\) the extremals \(u_\varepsilon\) satisfy the corresponding Euler-Lagrange equation \(-\Delta u_\varepsilon=\lambda_\varepsilon f(u_\varepsilon)\) in \(\Omega, u_\varepsilon=0\) on \(\partial\Omega\) with \(f=F'\). The main result says 1) The extremals \((u_\varepsilon)\) concentrate at a single point \(x_0 \in \overline{\Omega}\). 2) On a microscopic scale near the concentration point they tend to an entire extremal, i.e. to a solution of the variational problem on \({\mathbb R}^n\) with \(\varepsilon = 1\). These results extend some work by P. L. Lions, using a new proof based on a local generalized Sobolev inequality. The identification of the concentration point \(x_0\) will be discussed in a forthcoming paper.

MSC:

35J20 Variational methods for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Bandle, C.; Flucher, M., Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations \(βU = e^U\) and \(βU = U^{(n+2)/(n-2)} \), SIAM Rev., Vol. 38, 2, 191-238 (1996) · Zbl 0857.35034
[2] Dellacherie, C.; Meyer, P.-A., (Probabilities and potential. Vol. 29 of North-Holland Mathematics Studies (1978), North-Holland Publishing Co: North-Holland Publishing Co Amsterdam)
[3] Flucher, M., An asymptotic formula for the minimal capacity among sets of equal area, Calc. Var. Partial Differential Equations, Vol. 1, 1, 71-86 (1993) · Zbl 0801.35153
[4] Flucher, M.; Müller, S., Radial symmetry and decay rate of variational ground states in the zero mass case, SIAM J. Math. Anal., Vol. 29, 3, 712-719 (1998) · Zbl 0908.35005
[5] M. Flucher, A. GarroniS. Müller; M. Flucher, A. GarroniS. Müller
[6] Flucher, M.; Rumpf, M., Bernoulli’s free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math., Vol. 486, 165-204 (1997) · Zbl 0909.35154
[7] Flucher, M.; Wei, J., Asymptotic shape and location of small cores in elliptic free-boundary problems, Math. Z, Vol. 228, 683-703 (1998) · Zbl 0921.35024
[8] (Gruber, P. M.; Wills, J. M., Handbook of convex geometry, Vol. B (1993), North-Holland Publishing Co: North-Holland Publishing Co Amsterdam) · Zbl 0777.52002
[9] Lions, P.-L., The concentration-compactness principle in the calculus of variations, (The locally compact case. I. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, Vol. 1 (1984)), 109-145, 2 · Zbl 0704.49005
[10] Lions, P.-L., The concentration-compactness principle in the calculus of variations, (The limit case. I. The limit case. I, Rev. Mat. Iberoamericana, Vol. 1 (1985)), 145-201, 1 · Zbl 0704.49005
[11] Morrey, Charles B., (Multiple integrals in the calculus of variations, Band 130 (1966), Springer-Verlag New York, Inc: Springer-Verlag New York, Inc New York), Die Grundlehren der mathematischen Wissenschaften · Zbl 0142.38701
[12] Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. 4, 3, 4, 697-718 (1976) · Zbl 0341.35031
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