Weighted fourth order equation of Kirchhoff type in dimension 4 with non-linear exponential growth
DOI:
https://doi.org/10.12775/TMNA.2023.005Keywords
Kirchhoff-Schrödinger equation, Adams' inequality, nonlinearity of exponential growth, mountain pass method, compactness levelAbstract
In this work, we are concerned with the existence of a ground state solution for a Kirchhoff weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb{R}^{4}$. The nonlinearities have critical growth in view of Adams' inequalities. To prove the existence result, we use Pass Mountain Theorem. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. 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.References
D.R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385–398.
F.S.B. Albuquerque, A. Bahrouni and U. Severo, Existence of solutions for a nonhomogeneous Kirchhoff–Schrödinger type equation in R2 involving unbounded or decaying potentials, Topol. Methods Nonlinear Anal. 56 (2020), no. 1, 263–281, DOI: 10.12775/TMNA.2020.013.
C.O. Alves, F.J.S.A. Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl. 49 (2005), 85–93.
C.O. Alves and F.J.S.A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal. 8 (2001), 43–56.
A. Ambrosetti and P.H. Rabionowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal.14 (1973), 349–381.
S. Baraket and R. Jaidane, Non-autonomous weighted elliptic equations with double exponential growth, An. Şt. Univ. ,,Ovidius” Constanţa 29 (2021), no. 3, 33–66.
M. Calanchi and B. Ruf, Trudinger–Moser type inequalities with logarithmic weights in dimension N , Nonlinear Anal. 121 (2015), 403-411, DOI: 10.1016/j.na.2015.02.001.
M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDea Nonlinear Differ. Equ. Appl. 24 (2017), Art. 29, DOI: 10.1007/s00030-017-0453-y.
B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl.394 (2012), 488–495.
S. Chen, X. Tang and J. Wei, Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), 38, DOI: 10.1007/s00033-020-01455-w.
M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619–4627.
M. Chipot and J.F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modélisation Mathématique et Analyse Numérique 26 (1992), 447–467.
S. Deng, T. Hu and C. Tang, N -Laplacian problems with critical double exponential nonlinearities, DIiscrete Contin .Syst. 41 (2021), 987–1003.
P. Drabek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, Berlin, 1997. DOI: 10.1515/9783110804775.
D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 139–153, DOI: 10.1007/BF01205003.
G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), no. 1, 23–39.
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1876.
M.K.-H. Kiessling, Statistical Mechanics of Classical Particles with Logarithmic Interactions, Communications on Pure and Applied Mathematics, vol. 46, 1993, pp. 27–56, DOI: 10.1002/cpa.3160460103.
J.L.Lions, On Some Questions in Boundary Value Problems of Mathematical Physics, North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 1978.
P.L. Lions, The concentration-compactness principle in the calculus of variations, Part 1, Rev. Mat. Iberoam. 11 (1985), 185–201.
B. Ruf and F. Sani, Sharp Adams-type inequalities in RN , Trans. Amer. Math. Soc. 365 (2013), no. 2, 645–670.
F. Sani, A biharmonic equation in R4 involving nonlinearities with critical exponential growth, Comm. Pure Appl. Anal. 12 (2013), no. 1, 405–428, DOI: 10.3934/cpaa.2013.12.405.
L. Wang and M. Zhu, Adams’ inequality with logarithm weight in R4 , Proc. Amer. Math. Soc. 149 (2021), no. 8, 3463–3472, DOI: org/10.1090/proc/15488.
C. Zhang, Concentration-compactness principle for Trudinger–Moser inequalities with logarithmic weights and their applications,4 Nonlinear Anal. 197 (2020), 1–22.
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