arXiv subjects

Masaki Sakuma

Publications and source records attributed to Masaki Sakuma.

4 recordsLinked to original sources

arXiv2024

Compact embedding from variable-order Sobolev space to $L^{q(x)}(Ω)$ and its application to Choquard equation with variable order and variable critical exponent

In this paper, we prove the compact embedding from the variable-order Sobolev space $W^{s(x,y),p(x,y)}_0 (Ω)$ to the Nakano space $L^{q(x)}(Ω)$ with a critical exponent $q(x)$ satisfying some conditions. It is noteworthy that the embedding can be compact even when $q(x)$ reaches the critical Sobolev exponent $p_s^*(x)$. As an application, we obtain a nontrivial solution of the Choquard equation \begin{equation*} \displaystyle (-Δ)_{p(\cdot,\cdot)}^{s(\cdot,\cdot)}u+|u|^{p(x,x)-2}u=\left(\int_Ω\frac{|u(y)|^{r(y)}}{|x-y|^{\frac{α(x)+α(y)}{2}}}dy\right) |u(x)|^{r(x)-2}u(x)\quad\text{in $Ω$} \end{equation*} with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition.

math.AP↗

arXiv2024

Infinitely many solutions for doubly critical variable-order $p(x)$-Choquard-Kirchhoff type equations via the concentration compactness method

In this paper, we prove the existence of infinitely many solutions of a doubly critical Choquard-Kirchhoff type equation \begin{equation*} \begin{split} &M(\mathcal{E}[u])(-Δ)_{p(\cdot,\cdot)}^{s(\cdot,\cdot)}u+V(x)|u|^{p(x)-2}u-\varepsilon_W W(x)|u|^{q_W(x)-2}u \\ &\quad\quad=\left(\int_{\mathbb{R}^N} \frac{|u(y)|^{r(y)}}{|x-y|^{α(x,y)}}dy\right) |u|^{r(x)-2}u + |u|^{q_c(x)-2}u\quad\text{in $\mathbb{R}^N$} \end{split} \end{equation*} with variable smoothness. The exponents $r(\cdot)$ and $q_c(\cdot)$ are allowed to have critical growths at the same time in the sense of Hardy-Littlewood-Sobolev inequality and Sobolev inequality, respectively. In the course of consideration, we formulate a new Sobolev type embedding theorem for the Slobodeckij spaces with variable smoothness and extend the concentration compactness lemma to the variable exponent case in the form involving some nonlocal quantities. To obtain a sequence of solutions, we use a critical point theorem based on Krasnoselskii's genus.

math.AP↗

arXiv2023

Ground states for $p$-fractional Choquard-type equations with critical local nonlinearity and doubly critical nonlocality

We consider a $p$-fractional Choquard-type equation \[ (-Δ)_p^s u+a|u|^{p-2}u=b(K\ast F(u))F'(u)+\varepsilon_g |u|^{p_g-2}u \quad\text{in $\mathbb{R}^N$}, \] where $0<s<1<p<p_g\leq p_s^*$, $N \geq \max\{2ps+α,p^2 s\}$, $a,b,\varepsilon_g\in (0,\infty)$, $K(x)= |x|^{-(N-α)}$, $α\in (0,N)$, and $F$ is a doubly critical nonlinearity in the sense of the Hardy-Littlewood-Sobolev inequality. It is noteworthy that the local nonlinearity may also have critical growth. Combining Brezis-Nirenberg's method with some new ideas, we obtain the ground state solutions via the mountain pass lemma and a generalized Lions-type theorem.

math.AP↗

arXiv2023

Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method

We prove the existence of infinitely many solutions to a fractional Choquard type equation \[ (-Δ)^s_p u+V(x)|u|^{p-2}u=(K\ast g(u))g'(u)+\varepsilon_W W(x)f'(u)\quad\text{in }\mathbb{R}^N \] involving fractional $p$-Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives a sequence of critical values converging to zero for even functionals. To assure the $(PS)_c$ conditions, we also use a nonlocal version of the concentration compactness lemma.

math.AP↗