Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 5 / October 2023
- Published online by Cambridge University Press:
- 31 October 2022, pp. 1683-1705
- Print publication:
- October 2023
-
- Article
- Export citation
-
In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel(0.1)
\begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation}for any nonnegative functions f\in L^{p}(\partial \mathbb {R}_+^{n}), g\in L^{q'}(\mathbb {R}_+^{n}) and p,\,\ q'\in (1,\,\infty ), \beta \geq 0, \alpha +\beta >1 such that \frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1.We prove the existence of all extremal functions for (0.1). We show that if f and g are extremal functions for (0.1) then both of f and g are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.
9 - Varopoulos Semigroups
-
- Book:
- Semigroups of Linear Operators
- Published online:
- 27 July 2019
- Print publication:
- 15 August 2019, pp 166-178
-
- Chapter
- Export citation
Groundstates for Choquard type equations with Hardy–Littlewood–Sobolev lower critical exponent
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 29 January 2019, pp. 1377-1400
- Print publication:
- June 2020
-
- Article
- Export citation
-
For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,
-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N,where N\ges 3, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and I_\alpha : {\open R}^N \to 0 is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided N \ges 4 or in dimension N = 3 if in addition 3/2 < α < 3 and \ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} , namely in the non-resonant case.
Existence of solutions for critical Choquard equations via the concentration-compactness method
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 26 January 2019, pp. 921-954
- Print publication:
- April 2020
-
- Article
- Export citation
-
In this paper, we consider the nonlinear Choquard equation
-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N,where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and G(u)=\int ^u_0g(s)\,{\rm d}s. Firstly, by assuming that the potential V(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.
