Ground state solutions for nonlinear Choquard equation with singular potential and critical exponents. (English) Zbl 1480.35223
Summary: In this paper, we study the ground state solutions to the following Choquard equation involving singular potential:
\[
-\Delta u+V(|x|) u = (I_{\alpha} \ast F(u)) f(u),\; x \in \mathbb{R}^N,
\]
where \(N \geqslant 3\), \(\alpha \in (0, N)\), \(I_{\alpha}\) is the Riesz potential, \(V\) is a singular potential with parameter \(\theta \in (\max \{ 0,4-N\}, 2)\cup (2,2N - 2) \cup (2N - 2, \infty)\), \(F\) is the primitive of \(f\) and \(f\) satisfies critical growth in sense of the Hardy-Littlewood-Sobolev inequality. Under different range of \(\theta\) and almost necessary conditions on the nonlinearity \(f\) in the spirit of Berestycki-Lions type conditions, we divide this paper into three parts. By virtue of two different kinds of Lions-type theorem and Nehari manifold, some existence results are established.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35J20 | Variational methods for second-order elliptic equations |
Keywords:
nonlinear Choquard equation; singular potential; critical expont; existence of ground statesReferences:
| [1] | Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 2, 277-320 (2006) · Zbl 1126.35057 |
| [2] | Badiale, M.; Rolando, S., A note on nonlinear elliptic problems with singular potentials, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., 17, 1, 1-13 (2006) · Zbl 1126.35021 |
| [3] | Badiale, M.; Benci, V.; Rolando, S., A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9, 3, 355-381 (2007) · Zbl 1149.35033 |
| [4] | Badiale, M.; Guida, M.; Rolando, S., Compactness and existence results for the p-Laplace equation, J. Math. Anal. Appl., 451, 1, 345-370 (2017) · Zbl 1368.35146 |
| [5] | Bartsch, T.; Ding, Y., On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313, 1, 15-37 (1999) · Zbl 0927.35103 |
| [6] | Battaglia, L.; Van Schaftingen, J., Groundstates of the Choquard equations with a sign-changing self-interaction potential, Z. Angew. Math. Phys., 69, 3, Article 86 pp. (2018), 16 pp. · Zbl 1395.35096 |
| [7] | Berestycki, H.; Lions, P., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 4, 313-345 (1983) · Zbl 0533.35029 |
| [8] | Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88, 3, 486-490 (1983) · Zbl 0526.46037 |
| [9] | Carrião, P.; Demarque, R.; Miyagaki, O., Nonlinear biharmonic problems with singular potentials, Commun. Pure Appl. Anal., 13, 6, 2141-2154 (2014) · Zbl 1304.35256 |
| [10] | Gao, F.; Yang, M., The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61, 7, 1219-1242 (2018) · Zbl 1397.35087 |
| [11] | Guo, L.; Hu, T.; Peng, S.; Shuai, W., Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Calc. Var. Partial Differ. Equ., 58, 4, Article 128 pp. (2019), 34 pp. · Zbl 1422.35077 |
| [12] | Li, Y.; Li, G.; Tang, C., Ground state solutions for Choquard equations with Hardy-Littlewood-Sobolev upper critical growth and potential vanishing at infinity, J. Math. Anal. Appl., 484, 2, Article 123733 pp. (2020), 15 pp. · Zbl 1433.35099 |
| [13] | Lieb, E., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., 57, 2, 93-105 (1976/77) · Zbl 0369.35022 |
| [14] | Lieb, E.; Loss, M., Analysis, Graduate Studies in Mathematics, vol. 14 (1997), American Mathematical Society: American Mathematical Society Providence, RI |
| [15] | Moroz, V.; Van Schaftingen, J., Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 2, 153-184 (2013) · Zbl 1285.35048 |
| [16] | Moroz, V.; Van Schaftingen, J., Existence of groundstates for a class of nonlinear Choquard equations, Trans. Am. Math. Soc., 367, 9, 6557-6579 (2015) · Zbl 1325.35052 |
| [17] | Moroz, V.; Van Schaftingen, J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52, 1-2, 199-235 (2015) · Zbl 1309.35029 |
| [18] | Panda, A.; Choudhuri, D.; Saoudi, K., A critical fractional Choquard problem involving a singular nonlinearity and a radon measure, J. Pseudo-Differ. Oper. Appl., 12, 1, Article 22 pp. (2021) · Zbl 1481.35199 |
| [19] | Pekar, S., Untersuchung ber die Elektronentheorie der Kristalle (1954), Akademie Verlag: Akademie Verlag Berlin · Zbl 0058.45503 |
| [20] | Penrose, R., On gravity’s role in quantum state reduction, Gen. Relativ. Gravit., 28, 5, 581-600 (1996) · Zbl 0855.53046 |
| [21] | Ruiz, D.; Van Schaftingen, J., Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differ. Equ., 264, 2, 1231-1262 (2018) · Zbl 1377.35011 |
| [22] | Strauss, A., Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55, 2, 149-162 (1977) · Zbl 0356.35028 |
| [23] | Su, J.; Wang, Z.; Willem, M., Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9, 4, 571-583 (2007) · Zbl 1141.35056 |
| [24] | Su, Y., Positive solution to Schrödinger equation with singular potential and double critical exponents, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., 31, 4, 667-698 (2020) · Zbl 1468.35042 |
| [25] | Willem, M., Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24 (1996), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0856.49001 |
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