A bifurcation problem associated to an asymptotically linear function

Sâanouni, Soumaya; Trabelsi, Nihed

doi:10.1016/s0252-9602(16)30102-3

Cited by 4 publications

(4 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…u is the density and div(a(x)∇u) represents the substance of diffusion through the system and finally f models the interaction of substances. In the stationary case and when f (x, t) = f (t) depends only on t and f is asymptotically linear at +∞, that is q(x) ≡ l = const., the problem (1) was studied by Sâanouni and Trabelsi in [21]. In fact the results in [21] was a generalisation of those founded by Mironescu and Rȃdulescu in [10,15,16,18] where a is constant with the same conditions on the nonlinearity f (t).…”

Section: Introduction and Main Resultsmentioning

confidence: 79%

“…In the stationary case and when f (x, t) = f (t) depends only on t and f is asymptotically linear at +∞, that is q(x) ≡ l = const., the problem (1) was studied by Sâanouni and Trabelsi in [21]. In fact the results in [21] was a generalisation of those founded by Mironescu and Rȃdulescu in [10,15,16,18] where a is constant with the same conditions on the nonlinearity f (t). Their proof of the existence of positive solutions is based on the condition f (0) > 0, since they take a positive, C 1 , convex increasing real values function f .…”

Section: Introduction and Main Resultsmentioning

confidence: 79%

See 1 more Smart Citation

Positive solutions for some asymptotically linear and superlinear weighted problems

Dammak

Zahed

Jerbi

2022

Filomat

View full text Add to dashboard Cite

In this paper, we study the following nonlinear elliptic problem -div(a(x) ?u) = f (x,u), x ? ? u ? H10(?) (P) where ? is a regular bounded domain in RN, N ? 2, a(x) a bounded positive function and the nonlinear reaction source is strongly asymptotically linear in the following sense lim t?+? f(x,t)/t = q(x) uniformly in x ? ?. We use a variant version of Mountain Pass Theorem to prove that the problem (P) has a positive solution for a large class of f (x,t) and q(x). Here, the existence of solution is proved without use neither the Ambrosetti-Rabionowitz condition nor one of its refinements. As a second result, we use the same techniques to prove the existence of solutions when f (x,t) is superlinear and subcritical on t at infinity.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 79%

Section: Introduction and Main Resultsmentioning

confidence: 79%

Positive solutions for some asymptotically linear and superlinear weighted problems

Dammak

Zahed

Jerbi

2022

Filomat

View full text Add to dashboard Cite

show abstract

“…When the function a(x) is a smooth on  and f(x, t) = g(t), with the same conditions (G1)-(G4), the problem (1.1) was studied by (Saanouni and Trabelsi, 2016a). The condition g(0) > 0 was capital in their work.…”

Section: Introduction With Main Resultsmentioning

confidence: 99%

Elliptic Weighted Problem with Indefinite Asymptotically Linear Nonlinearity

Zahed¹,

Alnaser²

2021

Journal of Mathematics and Statistics

View full text Add to dashboard Cite

The objective of this paper is the study the following nonlinear elliptic problem involving a weight function:-div(a(x)) = f(x, u) in  and u = 0 on  (P) where,  is a regular bounded subset N and N  2, a(x) is a nonnegative function and f(x, t) is allowed to be sign-changing. We employ variational techniques to prove the existence of a nontrivial solution for the problem (P), under some suitable assumptions, when the nonlinearity is asymptotically linear. Then, we prove by the same method the existence of positive solution when the function f is superlinear and subcritical at infinity.

show abstract

“…where, u is the density, div(a(x)u) represents the substance of diffusion through the system and f models the interaction of substances. In the stationary case and when f(x, s) depends only on s and f is asymptotically linear at +, the problem (1.1) was studied by (Saanouni and Trabelsi, 2016a). They considered the problem:…”

Section: Introductionmentioning

confidence: 99%