On the extremal solutions of superlinear Helmholtz problems

Makkia Dammak,Saber Ali Kharrati,Majdi El Ghord

doi:10.5269/bspm.42087

Abstract

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.

Highlights

On the Extremal Solutions of Superlinear Helmholtz ProblemsWe prove the existence of critical value λ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense
∞ and f is a positive, convexe and C1 function defined on [0, ∞)
We prove the existence of critical value λ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense

Summary

On the Extremal Solutions of Superlinear Helmholtz Problems

We prove the existence of critical value λ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. 2. For λ = λ∗, the problem (1.4) admits a unique weak solution u∗, u∗ = lim uλ, called the extremal λ−→λ∗. Given g ∈ L1(Ω), there exists a unique v ∈ L1(Ω) which is a weak solution of. Let v1 and v2 be two solutions of problem (2.1), v = v1 − v2 satisfies v(−∆ζ + cζ) = 0, Given any φ ∈ D(Ω), let ζ be solution of. Suppose that f is a function satisfies (1.3) and let u be a weak super solution of (1.4), there exists a weak solution u of the problem (1.4) with 0 ≤ u ≤ u.

We define now

This implies

For any

As λ