Positive and sign changing solutions to a nonlinear Choquard equation

By M. Clapp, Dora Salazar. 2012

148 citations recorded by the source corpus.

We consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$ is continuous, $\inf_{\mathbb{R}^{N}}W>0,$ and $W(x)$ tends to a positive constant as $|x|$ tends to infinity. Under symmetry assumptions on $\Omega$ and $W$, which allow finite symmetries, and some assumptions on the decay of $W$ at infinity, we establish the existence of a positive solution and multiple sign changing solutions to this problem, having small energy.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f829c8896b4ad5712f31412a308b38ba0966321e

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Citations in the corpus

On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents (2016, background): “int theorem; [35] for the existence of ground states under the assumptions of Berestycki-Lions type; [3] for the existence of multibump shaped solution for the equation with deepening potential well; [20, 26] for the existence of sign-changing solutions; [4, 5, 36, 41] for the existence and concentr”
The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations (2015, background): “Recently, Clapp and Salazar in [11] obtained radially symmetric sign-changing solutions for (1.”
Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations (2020, background): “For the Choquard-type equation and related problems, we refer to [11,31,33,44] for the existence of solutions and multiplicity properties, to [13,22,47] for existence of sign-changing solutions, and to [12,32,45,48] for semiclassical solutions.”
Least action nodal solutions for the quadratic Choquard equation (2015, background): “2 MARCO GHIMENTI, VITALY MOROZ, AND JEAN VAN SCHAFTINGEN with a parameter p > 1 attracted interest of mathematicians, see [3, 7, 8, 10,23,26] and further references therein; see also [9, 11] for modifications of (Cp) involving fractional Laplacian.”
Multi-bump solutions for Choquard equation with deepening potential well (2015, background): “For other related results, we refer the readers to [11, 18] for the existence of sign-changing solutions, [28, 34, 37] for the existence and concentration behavior of the semiclassical solutions and [29] for the critical nonlocal part with respect to the Hardy-Littlewood-Sobolev inequality.”
The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation (2016, background): “For other related results, we refer the readers to [14, 17] for the existence of sign-changing solutions, [4, 5, 27, 32, 36, 39] for the existence and concentration behavior of the semiclassical solutions.”
Semi-classical states for the Choquard equation (2013, background): “Recently, for ﬁxed ε > 0 and when V converges exponentially to a positive limit at inﬁnity, the existence at least one positive solution to ( ε ) has been proved [16, theorem 1.2; 20].”