Interpolating estimates with applications to some quantitative symmetry results

<abstract><p>We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.</p></abstract>

Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

In this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , where $\lambda >0$ λ > 0 is a parameter, $\alpha \in (0,N)$ α ∈ ( 0 , N ) , $N\ge 3$ N ≥ 3 , $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ I α : R N → R is the Riesz potential. As usual, $\alpha /N+1$ α / N + 1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$ λ > λ ∗ for some given number $\lambda _{*}$ λ ∗ in three cases: (i) $2< p<\frac{4}{N}+2$ 2 < p < 4 N + 2 , (ii) $p=\frac{4}{N}+2$ p = 4 N + 2 , and (iii) $\frac{4}{N}+2< p<2^{*}$ 4 N + 2 < p < 2 ∗ . Our result improves the previous related ones in the literature.

Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

Singularly perturbed quasilinear Choquard equations with nonlinearity satisfying Berestycki–Lions assumptions

In the present paper, we consider the following singularly perturbed problem: $$ \textstyle\begin{cases} -\varepsilon ^{2}\Delta u+V(x)u-\varepsilon ^{2}\Delta (u^{2})u= \varepsilon ^{-\alpha }(I_{\alpha }*G(u))g(u), \quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$ { − ε 2 Δ u + V ( x ) u − ε 2 Δ ( u 2 ) u = ε − α ( I α ∗ G ( u ) ) g ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $\varepsilon >0$ ε > 0 is a parameter, $N\ge 3$ N ≥ 3 , $\alpha \in (0, N)$ α ∈ ( 0 , N ) , $G(t)=\int _{0}^{t}g(s)\,\mathrm{d}s$ G ( t ) = ∫ 0 t g ( s ) d s , $I_{\alpha }: \mathbb{R}^{N}\rightarrow \mathbb{R}$ I α : R N → R is the Riesz potential, and $V\in \mathcal{C}(\mathbb{R}^{N}, \mathbb{R})$ V ∈ C ( R N , R ) with $0<\min_{x\in \mathbb{R}^{N}}V(x)< \lim_{|y|\to \infty }V(y)$ 0 < min x ∈ R N V ( x ) < lim | y | → ∞ V ( y ) . Under the general Berestycki–Lions assumptions on g, we prove that there exists a constant $\varepsilon _{0}>0$ ε 0 > 0 determined by V and g such that for $\varepsilon \in (0,\varepsilon _{0}]$ ε ∈ ( 0 , ε 0 ] the above problem admits a semiclassical ground state solution $\hat{u}_{\varepsilon }$ u ˆ ε with exponential decay at infinity. We also study the asymptotic behavior of $\{\hat{u}_{\varepsilon }\}$ { u ˆ ε } as $\varepsilon \to 0$ ε → 0 .

Weighted Estimates for Iterated Commutators of Riesz Potential on Homogeneous Groups

In this paper, we study the two weight commutators theorem of Riesz potential on an arbitrary homogeneous group H of dimension N. Moreover, in accordance with the results in the Euclidean space, we acquire the quantitative weighted bound on homogeneous group.

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

Existence and nonexistence in the liquid drop model

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

The Existence of Nontrivial Solution for a Class of Kirchhoff-Type Equation of General Convolution Nonlinearity without Any Growth Conditions

In this paper, we consider the following Kirchhoff-type equation:{u∈H1(RN),−(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα*F(u))f(u)+λg(u),inRN, where a>0, b≥0, λ>0, α∈(N−2,N), N≥3, V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N−2,N). Under certain assumptions on V(x), f(u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods.