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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

Citations:

Zbl 1412.35209
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References:

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