The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent

Divya Goel

DOI: http://dx.doi.org/10.12775/TMNA.2019.068

Abstract


In this article we are concerned with the following Choquard equation \begin{alignat*}2 -\Delta u &= \la|u|^{q-2}u +\bigg(\int_{\Omega}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\bigg)|u|^{2^*_{\mu}-2}u,\quad u> 0, &\quad& \text{in }\Omega,\\ u& = 0 &\quad& \text{on } \partial \Omega , \end{alignat*} where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^N( N\geq 3)$, $2^*_{\mu}=({2N-\mu})/({N-2})$ and $q \in [2,2^*)$ where $2^*={2N}/({N-2})$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\Omega$. Indeed, we prove that if $\la< \la_1$, then problem has $\text{cat}_{\Omega}(\Omega)$ positive solutions whenever $q \in [2,2^*)$ and $N> 3 $ or $4< q< 6 $ and $N=3$.

Keywords


Choquard equation; critical exponent; Lusternik-Schnirelman theory

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References


A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. Fr. Sér. 2 49, 1992.

C.O. Alves and Y.H. Ding, Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity, J. Math. Anal. Appl. 279 (2003), no. 2, 508–521.

C.O. Alves, G.M. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Adv. Nonlinear Anal. 5 (2016), 331–345.

A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294.

V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Ration. Mech. Anal. 114 (1991), 79–93.

V. Benci, G. Cerami and D. Passaseo, On the number of the positive solutions of some nonlinear elliptic problems, Nonlinear Analysis, A tribute in honour of G. Prodi, Quaderno Scuola Norm. Sup., Pisa, 1991, pp. 93–107.

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.

H. Brezis and E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233–248.

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl. 407 (2013), 1–5.

E.N. Dancer, A note on an equation with critical exponent, Bull. Lond. Math. Soc. 20 (1988), 600–602.

P. Drabek and S.I. Pohozaev, Positive solutions for the p-Laplacian: application of the Fibering method,, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 703–726.

G.M Figueiredo and G. Siciliano, Positive positive solutions for the fractional Laplacian in the almost critical case in a bounded domain, Nonlinear Anal. Real World Appl. 36 (2017), 89–100.

G.M Figueiredo, G.M. Bisci and R. Servadei, The effect of the domain topology on the number of solutions of fractional Laplace problems, Calc. Var. 57 (2018), 103.

F. Gao and M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), 1006–1041.

F. Gao and M. Yang, On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math. DOI: 10.1007/s11425-016-9067-5.

F. Gao, E.D. da Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, arXiv:1712.08264.

M. Ghimenti and D. Pagliardini, Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains, arXiv:1804.03448 (2018).

J. Giacomoni, T. Mukherjee and K. Sreenadh, Doubly nonlocal system with Hardy–Littlewood–Sobolev critical nonlinearity, J. Math. Anal. Appl. 467 (2018), 638–672.

J. Giacomoni, D. Goel, and K. Sreenadh, Regularity results on a class of doubly nonlocal problems (2019), arXiv:1909.10648.

D. Goel, V. Rădulescu and K. Sreenadh, Coron problem for nonlocal equations invloving Choquard nonlinearity, DOI: 10.1515/ans-2019-2064.

E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1976/77), 93–105.

E.H. Lieb and M. Loss, Analysis , Graduate Studies in Mathematics, Vol. 14, Amer. Math. Soc., Providence, Rhode Island, 2001.

V. Moroz, J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579.

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), 1550005.

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal. 13 (1989), 1241–1249.

G. Siciliano, Multiple positive solutions for a Schrödinger–Poisson–Slater system, J. Math. Anal. Appl. 365 (2010), no. 1, 288–299.

M. Struwe, Variational Methods, Springer, New York, 1990.

M. Willem, Minimax Theorems, Birkhäuser, Boston, MA, 1996.

H. Yin and Z. Yang, Multiplicity of positive solutions to a (p, q) Laplacian equation involving critical nonlinearity, Nonlinear Anal. 75 (2012), 3021–3035.


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