Geodesic
Weighted Schrödinger-Kirchhoff type problem in dimension 2 with non-linear double exponential growth
Filomat, Tome 37 (2023) no. 16, p. 5373

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In this work, we study the weighted Kirchhoff problem1 ( ∫ B (σ(x)|∇u|2 + V(x)u2) dx )[ − div(σ(x)∇u) + V(x)u ] = f (x,u) in B u > 0 in B u = 0 on ∂B, where B is the unit ball in R2, σ(x) = log e |x| , the singular logarithm weight in the Trudinger-Moser embedding, 1 is a continuous positive function on R+ and the potential V is a continuous positve function. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities. We prove the existence of non-trivial solutions via the critical point theory. In the critical case, the associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.
DOI : 10.2298/FIL2316373B
Classification : 35J20, 35J30, 35K57, 35J60
Keywords: Kirchhoff-Schrödinger equation, Moser-Trudinger’s inequality, Nonlinearity of double exponential growth, Mountain pass method, Compactness level 
Sami Baraket; Rached Jaidane. Weighted Schrödinger-Kirchhoff type problem in dimension 2 with non-linear double exponential growth. Filomat, Tome 37 (2023) no. 16, p. 5373 . doi: 10.2298/FIL2316373B
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     author = {Sami Baraket and Rached Jaidane},
     title = {Weighted {Schr\"odinger-Kirchhoff} type problem in dimension 2 with non-linear double exponential growth},
     journal = {Filomat},
     pages = {5373 },
     year = {2023},
     volume = {37},
     number = {16},
     doi = {10.2298/FIL2316373B},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2316373B/}
}
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