A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators
DOI:
https://doi.org/10.12775/TMNA.2023.037Keywords
Critical elliptic problems, differences of local and nonlocal operators, multiplicity of solutionsAbstract
We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative energy and the other with positive energy, for all sufficiently small values of a parameter. The proof is based on an abstract result recently obtained in \cite{MR4293883}.References
C.O. Alves, V. Ambrosio and T. Isernia, Existence, multiplicity and concentration for a class of fractional p&q Laplacian problems in RN , Commun. Pure Appl. Anal. 18 (2019), 2009–2045; English transl.: Birkhäuser, 1985.
V. Ambrosio and T. Isernia, On a fractional p&q Laplacian problem with critical Sobolev–Hardy exponents, Mediterr. J. Math. 15 (2018), paper no. 219, 17 pp.
V. Ambrosio, T. Isernia and G. Siciliano, On a fractional p&q Laplacian problem with critical growth, Minimax Theory Appl. 4 (2019), 1–19.
G. Arioli and F. Gazzola, Some results on p-Laplace equations with a critical growth term, Differential Integral Equations 11 (1998), 311–326.
M. Bhakta and D. Mukherjee, Multiplicity results for (p, q) fractional elliptic equations involving critical nonlinearities, Adv. Differential Equations 24 (2019), 185–228.
S. Biagi, E. Vecchi, S. Dipierro and E. Valdinoci, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), 1611–1641.
L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Art. 23, 32.
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
F. Chen and Y. Yang, Existence of solutions for the fractional (p, q)-Laplacian problems involving a critical Sobolev exponent, Acta Math. Sci. Ser. B (Engl. Ed.) 40 (2020), 1666–1678.
M. Degiovanni and S. Lancelotti, Linking solutions for p-Laplace equations with nonlinearity at critical growth, J. Funct. Anal. 256 (2009), 3643–3659.
S. Dipierro, K. Perera, C. Sportelli and E. Valdinoci, An existence theory for superposition operators of mixed order subject to jumping nonlinearities, Nonlinearity 37 (2024), no. 5, DOI: 10.1088/1361-6544/ad3793.
S. Dipierro, K. Perera, C. Sportelli and E. Valdinoci, An existence theory for nonlinear superposition operator of mixed fractional order, ArXiv Preprint, posted on 2023, DOI: arXiv.2310.02628.
S. Dipierro and E. Valdinoci, Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes, Phys. A 575 (2021), paper no. 126052, 20.
H. Egnell, Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. Rational Mech. Anal. 104 (1988), 57–77.
E.R. Fadell and P.H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139–174.
J.P. Garcı́a Azorero and I. Peral Alonso, Existence and nonuniqueness for the pLaplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), 1389–1430.
D. Goel, D. Kumar and K. Sreenadh, Regularity and multiplicity results for fractional (p, q)-Laplacian equations, Commun. Contemp. Math. 22 (2020), 1950065, 37.
M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879–902.
P. Mironescu and W. Sickel, A Sobolev non embedding, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), 291–298.
S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Brezis–Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations 55 (2016), Art. 105, 25.
K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003), 301–309.
K. Perera, An abstract critical point theorem with applications to elliptic problems with combined nonlinearities, Calc. Var. Partial Differential Equations 60 (2021), paper no. 181, 23.
K. Perera, R.P. Agarwal and D. O’Regan, Morse theoretic aspects of p-Laplacian type operators, Math. Surveys Monogr. American, Mathematical Society, Providence, RI, 2010.
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Kanishka Perera, Caterina Sportelli
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
