Positive solutions to the sublinear Lane-Emden equation are isolated. (English) Zbl 1482.35150

The authors establish the following fine and important property of positive solutions of the sublinear Lane-Emden problem \(-\Delta u = |u|^{q-2} u\) in a bounded open set \(\Omega\) with zero Dirichlet boundary conditions on \(\partial \Omega\), \(1<q<2\). Namely, it is proved that, under certain regularity assumptions on \(\Omega\), the unique positive minimizer \(w\) of the energy functional \(\frac{1}{2} \int_\Omega |\nabla u|^2 \,dx - \frac{1}{2} \int_\Omega |u|^q \,dx\) in \(\mathcal{D}_0^{1,2}(\Omega)\) is isolated in the \(L^1(\Omega)\)-norm topology. As a consequence, the least generalized frequency \(\lambda_1 = \min\{\int_\Omega |\nabla u|^2 \,dx:~ \int_\Omega |u|^q \,dx=1\}\) is isolated in the sense that the second smallest frequency \(\lambda_2\) satisfies \(\lambda_1 < \lambda_2\). In contrast, if the required regularity assumptions on \(\Omega\) are violated, \(\lambda_1\) might not be isolated, see [L. Brasco and G. Franzina, Adv. Nonlinear Anal. 8, 707–714 (2019; Zbl 1412.35209)]. As the main auxiliary tool for the proof, the authors comprehensively study the weighted embedding \(\mathcal{D}_0^{1,2}(\Omega)\hookrightarrow L^2(\Omega;w^{q-2})\).


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35J25 Boundary value problems for second-order elliptic equations
35J61 Semilinear elliptic equations
35R09 Integro-partial differential equations
49R05 Variational methods for eigenvalues of operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces


Zbl 1412.35209
Full Text: DOI arXiv


[1] Deny, J.; Lions, J.-L., Les espaces de Beppo Levi, Ann. Inst. Fourier, 5, 305-370 (1954) · Zbl 0065.09903
[2] Brasco, L.; Franzina, G. (2019)
[3] Henrot, A. (2006). Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhauser Verlag, Basel. · Zbl 1109.35081
[4] Brasco, L.; Franzina, G., A pathological example in nonlinear spectral theory, Adv. Nonlinear Anal, 8, 1, 707-714 (2017) · Zbl 1412.35209
[5] Aronson, D. G.; Peletier, L. A., Large time behaviour of solutions of the porous medium equation in bounded domains, J. Diff. Equations, 39, 3, 378-412 (1981) · Zbl 0475.35059
[6] Vazquez, J. L., The Dirichlet problem for the porous medium equation in bounded domains, Asymptotic Behav. Monatsh. Math, 142, 1-2, 81-111 (2004) · Zbl 1055.35024
[7] Brasco, L., Volzone, B. (2020). Long-time behavior for the porous medium equation with small initial energy, preprint. Available at arxiv.org/abs/2011.04619.
[8] Benci, V.; Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal, 114, 1, 79-93 (1991) · Zbl 0727.35055
[9] Benci, V.; Cerami, G.; Passaseo, D., Nonlinear Analisys. A Tribute in Honour of G. Prodi, On the number of the positive solutions of some nonlinear elliptic problems, 93-109 (1991), Quaderno Scuola Normale Superiore, Pisa · Zbl 0838.35040
[10] Dancer, E. N., The effect of domain shape on the number of positive solutions of certain nonlinear equations, II, J. Diff. Equations, 87, 2, 316-339 (1990) · Zbl 0729.35050
[11] Dancer, E. N., The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Diff. Equations, 74, 1, 120-156 (1988) · Zbl 0662.34025
[12] Nazarov, A. I., The one-dimensional character of an extremum point of the Friedrichs inequality in spherical and plane layers, J Math Sci, 102, 5, 4473-4486 (2000)
[13] Dancer, E. N., On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann, 285, 4, 647-669 (1989) · Zbl 0699.35103
[14] Ercole, G., Sign-definiteness of \(####\) eigenfunctions for a super-linear \(####\) Laplacian eigenvalue problem, Arch. Math, 103, 189-194 (2014) · Zbl 1317.35165
[15] Damascelli, L.; Grossi, M.; Pacella, F., Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 5, 631-652 (1999) · Zbl 0935.35049
[16] Lin, C.-S., Uniqueness of least energy solutions to a semilinear elliptic equation in, Manuscripta Math, 84, 13-19 (1994) · Zbl 0807.35043
[17] Brasco, L.; Franzina, G.; Ruffini, B., Schrödinger operators with negative potentials and Lane-Emden densities, J. Funct. Anal, 274, 6, 1825-1863 (2018) · Zbl 1388.35135
[18] Carbery, A.; Maz’ya, V.; Mitrea, M.; Rule, D., The integrability of negative powers of the solution of the Saint Venant problem, Ann. Sc. Norm. Super. Pisa Cl. Sci, 13, 465-531 (2014) · Zbl 1297.35088
[19] Maeda, F.-Y.; Suzuki, N., The integrability of superharmonic functions on Lipschitz domains, Bull. London Math. Soc, 21, 3, 270-278 (1989) · Zbl 0645.31005
[20] Nečas, J., Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa., 16, 3, 305-326 (1962) · Zbl 0112.33101
[21] Kufner, A., Weighted Sobolev Spaces (1985), New York: A Wiley-Interscience Publication. John Wiley & Sons, Inc, New York · Zbl 0567.46009
[22] Brezis, H.; Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Anal, 10, 1, 55-64 (1986) · Zbl 0593.35045
[23] Maz’ya, V., Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Second, Revised and Augmented Edition, 342 (2011), Heidelberg: Springer, Heidelberg
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.