On the extremal solutions of superlinear Helmholtz problems
Keyword(s):
Abstract: In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.
2019 ◽
Vol 9
(1)
◽
pp. 1092-1101
◽
2018 ◽
Vol 20
(3)
◽
pp. 333-345
2021 ◽
Vol 31
(5)
◽
pp. 053120
2014 ◽
Vol 36
(1)
◽
pp. A1-A19
◽
Keyword(s):
Keyword(s):