Pohožaev-type ground state solutions for Choquard equation with singular potential and critical exponent
KeywordsChoquard equation, Lions-type theorem, Singular potential, Pohožaev-type ground state solution, Critical exponent
AbstractIn this paper, we are concerned with the existence of Pohožaev-type ground state solutions for the Choquard equation with a singular potential and a critical exponent. By virtue of a generalized version of the Lions-type theorem and the Pohožaev manifold, we obtain the existence of a Pohožaev-type ground state solution for the above problem. Some recent results from the literature are improved and extended.
M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006), no. 1, 1–13.
M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 6, Art. 67, 34.
M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the p-Laplace equation, J. Math. Anal. Appl. 451 (2017), no. 1, 345–370.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345.
H. Brézis and E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490.
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.
P.C. Carrião, R. Demarque and O.H. Miyagaki, Nonlinear biharmonic problems with singular potentials, Commun. Pure Appl. Anal. 13 (2014), no. 6, 2141–2154.
D. Cassani, and J.J. Zhang, Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth, Adv. Nonlinear Anal. 8 (2019), no. 1, 1184–1212.
F. Catrina, Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal. 3 (2014), no. 1, 1–13.
A.R. Li, J.B. Su and L.G. Zhao, Existence and multiplicity of solutions of Schrödinger–Poisson systems with radial potentials, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 2, 319–332.
A.R. Li and J.B. Su, Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in R3 , Z. Angew. Math. Phys. 66 (2015), no. 6, 3147–3158.
E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1976/77), no. 2, 93–105.
E.H. Lieb and B. Simon, The Hartree–Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), no. 3, 185–194.
E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997.
P.L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), no. 6, 1063–1072.
I.M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger–Newton equations, Classical Quantum Gravity, vol. 15, 1998, Topology of the Universe Conference (Cleveland, OH, 1997), pp. 2733–2742.
V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184.
V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579.
V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005, 12.
S. Pekar, Untersuchung ber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativity Gravitation 28 (1996), no. 5, 581–600.
D. Ruiz, and J. Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations 264 (2018), no. 2, 1231–1262.
J. Seok, Nonlinear Choquard equations: doubly critical case, Appl. Math. Lett. 76 (2018), 148–156.
Z.F. Shen, F.S. Gao and M.B. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst. 38 (2018), no. 7, 3567–3593.
A.W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162.
J.B. Su, Z.Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9 (2007), no. 4, 571–583.
Y. Su, Positive solution to Schrödinger equation with singular potential and double critical exponents, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31 (2020), no. 4, 667–698.
Y. Su, L. Wang, H.B. Chen and S.L. Liu, Multiplicity and concentration results for fractional choquard equation: Doubly critical case, Nonlinear Anal. 198 (2020), no. 1, Article 111872.
J. Van Schaftingen and J.K. Xia, Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl. 464 (2018), no. 2, 1184–1202.
M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
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