## 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 $$\mathcal{D}_0^{1,2}(\Omega)\hookrightarrow L^2(\Omega;w^{q-2})$$.

### MSC:

 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
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### References:

 [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
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