| Issue |
ESAIM: COCV
Volume 24, Number 2, April–June 2018
|
|
|---|---|---|
| Page(s) | 569 - 578 | |
| DOI | https://doi.org/10.1051/cocv/2017048 | |
| Published online | 26 January 2018 | |
The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞★,★★
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
denisa.stancu@yahoo.com
4
“Simion Stoilow” Institute of Mathematics of the Romanian Academy,
010702
Bucharest, Romania
5
Department of MathematicsUniversity of Pecs,
7624
Pécs, Hungary
csvarga@cs.ubbcluj.ro
a Corresponding author: mmihailes@yahoo.com
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 −Δp u = 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 solutionviscosity solution / nonlinear elliptic equations / asymptotic behavior / distance function to the boundary
Corresponding author: Mihai Mihăilescu, Department of Mathematics, University of Craiova, 200585 Craiova, Romania. E-mail: mmihailes@yahoo.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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