Ground state solutions and infinitely many solutions for a nonlinear Choquard equation
Abstract In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert...
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2021
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oai:doaj.org-article:e3843bdb58c5480da74a95615b3def4c2021-11-21T12:07:13ZGround state solutions and infinitely many solutions for a nonlinear Choquard equation10.1186/s13661-021-01573-y1687-2770https://doaj.org/article/e3843bdb58c5480da74a95615b3def4c2021-11-01T00:00:00Zhttps://doi.org/10.1186/s13661-021-01573-yhttps://doaj.org/toc/1687-2770Abstract In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ where N ≥ 3 $N\geq 3$ , 0 < μ < N $0<\mu <N$ , 2 N − μ N ≤ p < 2 N − μ N − 2 $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ , ∗ represents the convolution between two functions. We assume that the potential function V ( x ) $V(x)$ satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.Tianfang WangWen ZhangSpringerOpenarticleChoquard equationGround state solutionsGeometrically distinct solutionsAnalysisQA299.6-433ENBoundary Value Problems, Vol 2021, Iss 1, Pp 1-15 (2021) |
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Choquard equation Ground state solutions Geometrically distinct solutions Analysis QA299.6-433 |
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Choquard equation Ground state solutions Geometrically distinct solutions Analysis QA299.6-433 Tianfang Wang Wen Zhang Ground state solutions and infinitely many solutions for a nonlinear Choquard equation |
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Abstract In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ where N ≥ 3 $N\geq 3$ , 0 < μ < N $0<\mu <N$ , 2 N − μ N ≤ p < 2 N − μ N − 2 $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ , ∗ represents the convolution between two functions. We assume that the potential function V ( x ) $V(x)$ satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem. |
| format |
article |
| author |
Tianfang Wang Wen Zhang |
| author_facet |
Tianfang Wang Wen Zhang |
| author_sort |
Tianfang Wang |
| title |
Ground state solutions and infinitely many solutions for a nonlinear Choquard equation |
| title_short |
Ground state solutions and infinitely many solutions for a nonlinear Choquard equation |
| title_full |
Ground state solutions and infinitely many solutions for a nonlinear Choquard equation |
| title_fullStr |
Ground state solutions and infinitely many solutions for a nonlinear Choquard equation |
| title_full_unstemmed |
Ground state solutions and infinitely many solutions for a nonlinear Choquard equation |
| title_sort |
ground state solutions and infinitely many solutions for a nonlinear choquard equation |
| publisher |
SpringerOpen |
| publishDate |
2021 |
| url |
https://doaj.org/article/e3843bdb58c5480da74a95615b3def4c |
| work_keys_str_mv |
AT tianfangwang groundstatesolutionsandinfinitelymanysolutionsforanonlinearchoquardequation AT wenzhang groundstatesolutionsandinfinitelymanysolutionsforanonlinearchoquardequation |
| _version_ |
1718419216463298560 |