## Multi-bump solutions for Choquard equation with deepening potential well.(English)Zbl 1347.35097

This paper is concerned with the following Choquard equation $-\Delta u+ (\lambda a(x)+1) u = \left(\frac{1}{| x|^\mu} \ast | u|^p\right)| u|^{p-2}u \text{ in } \mathbb R^3, {(C)_\lambda}$ where $$\mu \in (0,3)$$, $$p\in (2, 6-\mu)$$ and $$0\leq a(x)\in C(\mathbb R^3)$$ with $$\Omega=\int(a^{-1}(0))$$ being non-empty bounded open set with smooth boundary $$\partial \Omega$$. Moreover, there exists $$M_0>0$$ such that $$| \{x\in \mathbb R^3: a(x)\leq M_0\}| <+\infty$$ and $$\Omega =\bigcup_{j=1}^k \Omega_j$$ with $$\mathrm{dist}(\Omega_i,\Omega_j)>0$$ if $$i\neq j$$. Under these conditions, by variational method, the authors prove that, there is a constant $$\lambda_0>0$$ such that for any non-empty subset $$\Gamma \subset \{1, \cdots, k\}$$ and $$\lambda \geq \lambda_0$$, the problem $$(C)_\lambda$$ has a positive solution $$u_\lambda$$. Furthermore, for any sequence $$\lambda_n \rightarrow \infty$$, passing to a subsequence, $$\{u_{\lambda_n}\}$$ converges strongly in $$H^1(\mathbb{R}^3)$$ to a function $$u$$ with $$u=0$$ outside $$\Omega_\Gamma = \bigcup_{j\in \Gamma} \Omega_j$$ and $$u|_{\Omega_\Gamma}$$ is a least energy solution for the following problem $-\Delta u +u =\left(\int_{\Omega_\Gamma}\frac{| u|^p}{| x-y|^\mu}dy\right) | u|^{p-2}u \text{ in } \Omega_\Gamma, u \in H_0^1(\Omega_\Gamma).$

### MSC:

 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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### References:

  Alves, CO, Existence of multi-bump solutions for a class of quasilinear problems, Adv. Nonlinear Stud., 6, 491-509, (2006) · Zbl 1184.35146  Alves, C.O., Nóbrega, A.B.: Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator. arXiv:1602.03112v1 (2016) · Zbl 1189.81061  Ackermann, N, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037  Bartsch, T; Wang, ZQ, Existence and multiplicity results for some superlinear elliptic problems on $${\mathbb{R}}^N$$, Comm. Part. Diff. Equ., 20, 1725-1741, (1995) · Zbl 0837.35043  Bartsch, T; Wang, ZQ, Multiple positive solutions for a nonlinear Schrödinger equations, Z. Angew. Math. Phys., 51, 366-384, (2000) · Zbl 0972.35145  Bartsch, T; Pankov, A; Wang, ZQ, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3, 549-569, (2001) · Zbl 1076.35037  Berestycki, H; Lions, PL, Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-346, (1983) · Zbl 0533.35029  Buffoni, B; Jeanjean, L; Stuart, CA, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Am. Math. Soc., 119, 179-186, (1993) · Zbl 0789.35052  Coti Zelati, V., Rabinowitz, R.: Homoclinic type solutions for semilinear elliptic PDE on $${\mathbb{R}}^n$$. Comm. Pure. Appl. Math. 45(10), 1217-1269 (1992) · Zbl 0785.35029  Clapp, M; Ding, YH, Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential, Diff. Int. Equ., 16, 981-992, (2003) · Zbl 1161.35385  Clapp, M; Salazar, D, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407, 1-15, (2013) · Zbl 1310.35114  Cingolani, S; Clapp, M; Secchi, S, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63, 233-248, (2012) · Zbl 1247.35141  Pino, M; Felmer, P, Multipeak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, 127-149, (1998) · Zbl 0901.35023  Pino, M; Felmer, P, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4, 121-137, (1996) · Zbl 0844.35032  Ding, Y; Tanaka, K, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manus. Math., 112, 109-135, (2003) · Zbl 1038.35114  Ding, Y; Szulkin, A, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differ. Equ., 29, 397-419, (2007) · Zbl 1119.35082  Floer, A; Weinstein, A, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal., 69, 397-408, (1986) · Zbl 0613.35076  Ghimenti, M., Van Schaftingen, J.: Nodal solutions for the Choquard equation. J. Funct. Anal. 271(1), 107-135 (2016) · Zbl 1345.35046  Ghimenti, M., Moroz, V., Van Schaftingen, J.: Least Action nodal solutions for ghe quadratic Choquard equation. arXiv:1511.04779v1 (2015) · Zbl 1355.35079  Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93-105 (1976/1977) · Zbl 0369.35022  Lieb, E., Loss, M.: Analysis. Gradute Studies in Mathematics, AMS, Providence, Rhode island (2001) · Zbl 1225.35091  Lions, PL, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072, (1980) · Zbl 0453.47042  Li, GB, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Math., 14, 27-36, (1989) · Zbl 0628.10041  Miranda, C, Un’ osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3, 5-7, (1940) · JFM 66.0217.01  Ma, L; Zhao, L, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467, (2010) · Zbl 1185.35260  Moroz, V; Schaftingen, J, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184, (2013) · Zbl 1285.35048  Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557-6579 (2015) · Zbl 1325.35052  Moroz, V; Schaftingen, J, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52, 199-235, (2015) · Zbl 1309.35029  Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math. 17(5), 1550005, 12 (2015) · Zbl 1326.35109  Pekar, S.: Untersuchungüber die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954) · Zbl 0058.45503  Penrose, R, On gravity’s role in quantum state reduction, Gen. Relativ. Gravitat., 28, 581-600, (1996) · Zbl 0855.53046  Stuart, CA; Zhou, HS, Global branch of solutions for non-linear Schrödinger equations with deepening potential well, Proc. Lond. Math. Soc., 92, 655-681, (2006) · Zbl 1225.35091  Sato, Y; Tanaka, K, Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells, Trans. Am. Math. Soc., 361, 6205-6253, (2009) · Zbl 1198.35261  Secchi, S.: A note on Schrödinger-Newton systems with decaying electric potential. Nonlinear Anal. 72, 3842-3856 (2010) · Zbl 1187.35254  Séré, E, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27-42, (1992) · Zbl 0725.58017  Wang, ZP; Zhou, HS, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc. (JEMS), 11, 545-573, (2009) · Zbl 1172.35073  Wei, J; Winter, M, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50, 012905, (2009) · Zbl 1189.81061  Willem, M.: Minimax Theorems, Birkhäuser (1996) · Zbl 0856.49001
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