Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

• Authors: GKIKAS Konstantinos T; NGUYEN Phuoc-Tai
• Year of publication: 2023
• Type: Article in Periodical
• Magazine / Source: Proceedings of the Royal Society of Edinburgh Section A: Mathematics
• MU Faculty or unit: Faculty of Science
• Citation: GKIKAS, Konstantinos T and Phuoc-Tai NGUYEN. Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge University Press, 2023, 58 pp. ISSN 0308-2105. Available from: https://dx.doi.org/10.1017/prm.2023.122.
• web: https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/semilinear-elliptic-equations-involving-power-nonlinearities-and-hardy-potentials-with-boundary-singularities/1FA7BF7DD7DB916CFDAC66C169C4
• Doi: http://dx.doi.org/10.1017/prm.2023.122
• Keywords: Hardy potentials; boundary singularities; capacities; critical exponents; removable singularity; Keller-Osserman estimates; 35J60; 35J75; 35J10; 35J66
• Description: Let ? ? RN (N ? 3) be a C2 bounded domain and ? ? ?? be a C2 compact submanifold without boundary, of dimension k, 0 ? k ? N - 1. We assume that ? = {0} if k = 0 and ? = ?? if k = N - 1. Let d?(x) = dist (x, ?) and Lµ = ? + µ d-?2, where µ ? R. We study boundary value problems (P±) -Lµu ± |u|p-1u = 0 in ? and trµ,?(u) = ? on ??, where p > 1, ? is a given measure on ?? and trµ,?(u) denotes the boundary trace of u associated to Lµ. Different critical exponents for the existence of a solution to (P±) appear according to concentration of ?. The solvability for problem (P+) was proved in [3, 29] in subcritical ranges for p, namely for p smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of -Lµ, we provide conditions on ? expressed in terms of capacities for the existence of a (unique) solution to (P+) in supercritical ranges for p, i.e. for p equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to (P-) under a smallness assumption on ?.