|Publication ahead of print|
|Published online||26 January 2018|
1 Department of Mathematics, University of Craiova, 200585 Craiova, Romania
2 Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania
3 Department of Mathematics and Computer Science, University Politehnica of Bucharest, 060042 Bucharest, Romania
4 “Simion Stoilow” Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania
5 Department of Mathematics, University of Pécs, 7624, Pécs, Hungary
Corresponding author: firstname.lastname@example.org
Received: 29 December 2016
Revised: 17 June 2017
Accepted: 19 June 2017
Let Ω ⊂ RN (N ≥ 2) be a bounded domain with smooth boundary. We show the existence of a positive real number λ⋆ such that for each λ ∈ (0,λ⋆) and each real number p > N the equation − Δpu = λeu in Ω subject to the homogeneous Dirichlet boundary condition possesses a nonnegative solution up. Next, we analyze the asymptotic behavior of up as p → ∞ and we show that it converges uniformly to the distance function to the boundary of the domain.
Mathematics Subject Classification: 35D30 / 35D40 / 35J60 / 47J30 / 46E30
Key words: Weak solution / viscosity solution / nonlinear elliptic equations / asymptotic behavior / distance function to the boundary
Correspondence address: Mihai Mihăilescu, Department of Mathematics, University of Craiova, 200585 Craiova, Romania. E-mail: email@example.com.
The research of M. Mihăilescu was partially supported by an UBB Advanced Fellowship-Intern granted by Star-UBB Institute, no. CNFIS-FDI-2016-0056. D. Stancu−Dumitru has been partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-PD-2016-0202. The research of C. Varga has been partially supported by OTKA (grant no. K 115926).
© EDP Sciences, SMAI 2018
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