Geodesic
Positive solutions for some asymptotically linear and superlinear weighted problems
Filomat, Tome 36 (2022) no. 1, p. 195

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DOI

In this paper, we study the following nonlinear elliptic problem −div(a(x)u) = f (x, u), x ∈ Ω u ∈ H 1 0 (Ω) (P) where Ω is a regular bounded domain in R N , 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 ∈ Ω. 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
DOI : 10.2298/FIL2201195D
Classification : 35J05, 35J65, 35J20, 35J60, 35K57, 35J70
Keywords: Asymptotically linear, mountain pass theorem, weighted problem, Cerami sequence 
Makkia Dammak; Hanadi Zahed; Chahira Jerbi. Positive solutions for some asymptotically linear and superlinear weighted problems. Filomat, Tome 36 (2022) no. 1, p. 195 . doi: 10.2298/FIL2201195D
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     title = {Positive solutions for some asymptotically linear and superlinear weighted problems},
     journal = {Filomat},
     pages = {195 },
     year = {2022},
     volume = {36},
     number = {1},
     doi = {10.2298/FIL2201195D},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2201195D/}
}
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