Moderate solutions of semilinear elliptic equations with Hardy potential

Marcus, Moshe; Nguyen, Phuoc-Tai

doi:10.1016/j.anihpc.2015.10.001

Marcus, Moshe ; Nguyen, Phuoc-Tai

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 69-88

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Résumé

Let Ω be a bounded smooth domain in $R^{N}$ . We study positive solutions of equation (E) $- L_{μ} u + u^{q} = 0$ in Ω where $L_{μ} = Δ + \frac{μ}{δ^{2}}$ , $0 < μ$ , $q > 1$ and $δ (x) = dist (x, \partial Ω)$ . A positive solution of (E) is moderate if it is dominated by an $L_{μ}$ -harmonic function. If $μ < C_{H} (Ω)$ (the Hardy constant for Ω) every positive $L_{μ}$ -harmonic function can be represented in terms of a finite measure on ∂Ω via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, $1 < q < q_{μ, c}$ . (The critical value depends only on N and μ.) For $q \geq q_{μ, c}$ there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator $L_{μ}$ . These results form the basis for the study of the nonlinear problem.

MR Zbl

DOI : 10.1016/j.anihpc.2015.10.001

Classification : 35J60, 35J75, 35J10
Keywords: Hardy potential, Martin kernel, Moderate solutions, Normalized boundary trace, Critical exponent, Removable singularities

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     author = {Marcus, Moshe and Nguyen, Phuoc-Tai},
     title = {Moderate solutions of semilinear elliptic equations with {Hardy} potential},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {69--88},
     publisher = {Elsevier},
     volume = {34},
     number = {1},
     year = {2017},
     doi = {10.1016/j.anihpc.2015.10.001},
     mrnumber = {3592679},
     zbl = {1356.35114},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.10.001/}
}

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Marcus, Moshe; Nguyen, Phuoc-Tai. Moderate solutions of semilinear elliptic equations with Hardy potential. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 69-88. doi: 10.1016/j.anihpc.2015.10.001

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