A note on the LpL^p-Sobolev inequality

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. This note is concerned, instead, with function restricted to bounded domain $\Omega\subset \mathbb{R}^N$. Based on the recent work of Figalli and Zhang [Duke Math. J., 2022], a remainder term with weak norm is established $$\frac{\|\nabla u\|_{L^p(\Omega)}}{\|u\|_{L^{p^*}(\Omega)}} -\mathcal{S} \geq \mathcal{C} \left(\frac{\|u\|_{L^{\bar{p}}_w(\Omega)}} {\|u\|_{L^{p^*}(\Omega)}}\right)^{\max\{2,p\}},\quad \forall u\in C^\infty_0(\Omega)\setminus\{0\},$$ for some $\mathcal{C}=\mathcal{C}(N,p,\Omega)>0$, where $\bar{p}=p^*(p-1)/p$ and $\|\cdot\|_{L^{\bar{p}}_w(\Omega)}$ denotes the weak $L^{\bar{p}}$-norm. Furthermore, the weak norm can not be replaced by the strong norm. This result answers the long-standing open problem raised by Bianchi and Egnell [J. Funct. Anal., 1991]

Existence of solution for a Kirchhoff type problem involving the fractional p-Laplace operator

This paper is concerned with the existence of solutions to a Kirchhoff type problem involving the fractional $p$-Laplacian operator. We obtain the existence of solutions by Ekeland's variational principle

Normalized solutions for a Choquard equation with exponential growth in R2\mathbb{R}^{2}

In this paper, we study the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} &-\Delta u+\lambda u=(I_{\alpha}\ast F(u))f(u), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$ is prescribed, $\lambda\in \mathbb{R}$, $\alpha\in(0,2)$, $I_{\alpha}$ denotes the Riesz potential, $\ast$ indicates the convolution operator, the function $f(t)$ has exponential growth in $\mathbb{R}^{2}$ and $F(t)=\int^{t}_{0}f(\tau)d\tau$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.Comment: arXiv admin note: text overlap with arXiv:2210.02331. text overlap with arXiv:2102.03001 by other author

Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities

Based on the work of Lin \cite{Li86}, a new second-order Caffarelli-Kohn-Nirenberg type inequality will be established, namely \begin{equation*} \int_{\mathbb{R}^N}|x|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N}|x|^{\beta}|u|^{p^*_{\alpha,\beta}} \mathrm{d}x\right)^{\frac{2}{p^*_{\alpha,\beta}}}, \quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N), \end{equation*} for some constant $\mathcal{S}>0$, where \begin{align*} N\geq 5,\quad \alpha>2-N,\quad \alpha-2<\beta\leq \frac{N}{N-2}\alpha,\quad p^*_{\alpha,\beta}=\frac{2(N+\beta)}{N-4+2\alpha-\beta}. \end{align*} We obtain a symmetry breaking conclusion: when $\alpha>0$ and $\beta_{\mathrm{FS}}(\alpha)<\beta< \frac{N}{N-2}\alpha$ where $\beta_{\mathrm{FS}}(\alpha):= -N+\sqrt{N^2+\alpha^2+2(N-2)\alpha}$, then the extremal function for the best constant $\mathcal{S}$, if it exists, is nonradial. This extends the works of Felli and Schneider \cite{FS03}, Ao, DelaTorre and Gonz\'{a}lez \cite{ADG22} to second-order case.Comment: arXiv admin note: text overlap with arXiv:2308.0601

Caffarelli-Kohn-Nirenberg-type inequalities related to weighted pp-Laplace equations

We use a suitable transform related to Sobolev inequality to investigate the sharp constants and optimizers for some Caffarelli-Kohn-Nirenberg-type inequalities which are related to the weighted $p$-Laplace equations. Moreover, we give the classification to the linearized problem related to the radial extremals. As an application, we investigate the gradient type remainder term of related inequality by using spectral estimate combined with a compactness argument which extends the work of Figalli and Zhang (Duke Math. J. 2022) at least for radial case.Comment: 39 pages, All comments are welcom

High energy sign-changing solutions for Coron's problem

We study the existence of sign changing solutions to the following problem {Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where [Formula presented] is the critical Sobolev exponent and Ωε is a bounded smooth domain in Rn, n≥3, of the form Ωε=Ω\B(0,ε). Here Ω is a smooth bounded domain containing the origin 0 and B(0,ε) denotes the ball centered at the origin with radius ε&gt;0. We construct a new type of sign-changing solutions with high energy to problem (0.1), when the parameter ε is small enough.</p

On a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth

In this paper, we study the following Hamiltonian Choquard-type elliptic systems involving singular weights \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\Delta u + V(x)u = \Big(I_{\mu_{1}}\ast \frac{G(v)}{|x|^{\alpha}}\Big)\frac{g(v)}{|x|^{\alpha}} \ \ \ & \mbox{in} \ \mathbb{R}^{2},\\[2mm] -\Delta v + V(x)v = \Big(I_{\mu_{2}}\ast \frac{F(u)}{|x|^{\beta}}\Big)\frac{f(u)}{|x|^{\beta}} \ \ \ & \mbox{in} \ \mathbb{R}^{2}, \end{array} \right. \end{aligned} \end{eqnarray*} where $\mu_{1},\mu_{2}\in(0,2)$, $0<\alpha \leq \frac{\mu_{1}}{2}$, $0<\beta \leq \frac{\mu_{2}}{2}$, $V(x)$ is a continuous positive potential, $I_{\mu_{1}}$ and $I_{\mu_{2}}$ denote the Riesz potential, $\ast$ indicates the convolution operator, $F(s),G(s)$ are the primitive of $f(s),g(s)$ with $f(s),g(s)$ have exponential growth in $\mathbb{R}^{2}$. Using the linking theorem and variational methods, we establish the existence of solutions to the above problem