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Olímpio H. Miyagaki

Publications and source records attributed to Olímpio H. Miyagaki.

14 recordsLinked to original sources
arXiv2026

Existence of solutions for elliptic problems involving the $(1,q)$-Laplacian operator and a discontinuous superlinear nonlinearity

In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(Ω)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $β\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.

math.AP↗
arXiv2025

Multiple nodal solutions of planar Stein-Weiss equations

In this paper, our goal is to investigate the existence of multiple nodal solutions to a class of planar Stein-Weiss problems involving a nonlinearity $f$ with subcritical or critical growth in the sense of Trudinger-Moser. To achieve this, we combine a gluing approach with the Nehari manifold argument. We demonstrate that for any positive integer $k\in \mathbb{N}$, the problem studied has at least one radially symmetrical ground state solution that changes sign exactly $k$-times.

math.AP↗
arXiv2024

Fractional Sobolev-Choquard critical systems with Hardy term and weighted singularities

In this paper, we consider a fractional p-Laplacian system of equations in the entire space RN with doubly critical singular nonlinearities involving a local critical Sobolev term together with a nonlocal Choquard critical term; the problem also includes a homogeneous singular Hardy term; moreover, all the nonlinearities involve singular critical weights. To prove the main result we use a refinement of Sobolev inequality that is related to Morrey space because our problem involves doubly critical exponentss and a version of the Caffarelli-Kohn-Nirenberg inequality. With the help of these results, we provide sufficient conditions under which a weak nontrivial solution to the problem exists via variational methods.

math.AP↗
arXiv2023

Fractional Sobolev-Chocard critical equation with Hardy term and weighted singularities

In this paper we consider a fractional $p$-Laplacian equation in the entire space $\mathbb{R}^{N}$ with doubly critical singular nonlinearities involving a local critical Sobolev term together with a nonlocal Choquard critical term; the problem also includes a homogeneous singular Hardy term. More precisely, we deal with the problem \begin{align*} \begin{cases} (-Δ)^{s}_{p,θ} u -γ\dfrac{|u|^{p-2}u}{|x|^{sp+ θ}} = \dfrac{|u|^{p^*_s(β,θ)-2}u}% {|x|^β} + \left[ I_μ \ast F_{δ,θ,μ}(\cdot, u) \right](x)f_{δ,θ,μ}(x,u) u \in \dot{W}^{s,p}_θ(\mathbb{R}^N) \end{cases} \end{align*} where $0 < s < 1$; $0 < α, \,β< sp + θ< N$; $0 < μ< N$; $2δ+ μ< N$; $γ< γ_{H}$ with the best fractional Hardy constant $γ_{H}$; the Hardy-Sobolev and Stein-Weiss upper critical fractional exponents are respectively defined by $p^*_s(β,θ) := p(N-β)/(N-sp-θ)$, and $p^\sharp_s(δ,θ,μ) := p(N-δ-μ/2)/(N-sp-θ)$. Moreover, $I_μ(x) =|x|^{-μ}$ is the Riesz potencial; $f_{δ,θ,μ}(x,t) := |x|^{-δ} |t|^{p^{\sharp}_{s}(δ,θ,μ)-2}t$ and $F_{δ,θ,μ}(x,t) := |x|^δ |t|^{p^{\sharp}_{s}(δ,θ,μ)}$; and the term with convolution integral is known as Choquard type nonlinearity. To prove the main result we have to show new embeddings involving the weighted Morrey spaces and a version of the Caffarelli-Kohn-Nirenberg inequality. With the help of these new embedding results, we provide sufficient conditions under which a weak nontrivial solution to the problem exists via variational methods.

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arXiv2021

Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity

In the present work we are concerned with the existence of normalized solutions to the following Schrödinger-Poisson System $$ \left\{ \begin{array}{ll} -Δu + λu + μ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ } \mathbb{R}^2 , \\ \intR |u(x)|^2 dx = c,\ c> 0 , \end{array} \right. $$ for $μ\in \R $ and a nonlinearity $f$ with exponential critical growth. Here $ λ\in \R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in \cite{Ji} and \cite{[cjj]}.

math.AP↗
arXiv2020

The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth

In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.

math.AP↗
arXiv2019

A fractional p-Laplacian problem with multiple critical Hardy-Sobolev nonlinearities

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities, \begin{align*} (-Δ_p)^su - μ\dfrac{\vert u \vert^{p-2} u}{\vert x \vert^{ps}} = \dfrac{\vert u\vert^{p^*_s(β)-2}u}{\vert x \vert^β} + \dfrac{\vert u\vert^{p^*_s(α)-2} u}{\vert x \vert^α}, \end{align*} where $ x \in \mathbb{R}^N$, $u\in D^{s,p}(\mathbb{R}^N)$, $0 sp$, $0<α 0$. To prove the existence of solution to the problem we have to formulate a refined version of the concentration-compactness principle and, as an independent result, we have to show that the extremals for the Sobolev inequality are attained.

math.AP↗
arXiv2019

Positive solutions for non-variational fractional elliptic systems with negative exponents

In this paper, we study strongly coupled elliptic systems in non-variational form with negative exponents involving fractional Laplace operators. We investigate the existence, nonexistence, and uniqueness of the positive classical solution. The results obtained here are a natural extension of the results obtained by Ghergu (2010), for the fractional case.

math.AP↗
arXiv2015

Concentration phenomena for fractional elliptic equations involving exponential critical growth

In this paper, we deal with the following singular perturbed fractional elliptic problem $ ε^{} (-Δ)^{1/2}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}, $ where $ (-Δ)^{1/2}u$ is the square root of the Laplacian and $f(s)$ has exponential critical growth. Under suitable conditions on $f(s)$, we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of $V$ as $ε$ goes to $0.$

math.AP↗
arXiv2015

Quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities: existence, regularity, nonexistence

This work deals with existence of solutions for the class of quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{u^{p-1}}{|y|^{p(a+1)}} = \frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} + \frac{u^{p^*(a,c)-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} The existence of a positive, weak solution $u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\})$ is proved with the help of the mountain pass theorem. We also prove a regularity result, that is, using Moser's iteration scheme we show that $u \in L_{\operatorname{loc}}^{\infty}(Ω)$ for domains $Ω\subset \mathbb{R}^{N-k}\times\mathbb{R}^{k}\backslash \{ |y|=0 \}$ not necessarily bounded. Finally we show that if $ u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\}) $ is a weak solution to the related problem \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{|u|^{p-2}u}{|y|^{p(a+1)}} = \frac{|u|^{q-2}u}{|y|^{bp^*(a,b)}} + \frac{|u|^{p^*(a,c)-2}u}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k, \end{align*} then $ u \equiv 0 $ when either $ 1 < q < p^*(a,b) $, or $ q > p^*(a,b) $ and $u \in L_{bp^*(a,b)/q, \operatorname{loc}}^{q} (\mathbb{R}^N\backslash\{|y|=0\}) \cap L_{\mathrm{loc}}^{\infty}(\mathbb{R}^{N-k}\times \mathbb{R}^{k} \backslash \{ |y| = 0\})$. This nonexistence of nontrivial solution is proved by using a Pohozaev-type identity.

math.AP↗
arXiv2015

Existence and multiplicity results on a class of quasilinear elliptic problems with cylindrical singularities involving multiple critical exponents

This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{u^{p-1}}{|y|^{p(a+1)}} = h\,\frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} +λg\,\frac{u^{q-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} We consider $N \geqslant 3$, $λ>0$, $p < k \leqslant N$, $1<p<N$, $0 \leqslant μ<\barμ \equiv \left\{[k-p(a+1)]/p\right\}^p$, $0 \leqslant a < (k-p)/p$, $a \leqslant b < a+1$, $a \leqslant c < a+1$, $1\leqslant q <p$, $p^*(a,b)=Np/[N-p(a+1-b)]$, and $p^*(a,c) \equiv Np/[N-p(a+1-c)]$; in particular, if $μ= 0$ we can include the cases $(k-p)/p \leqslant a < k(N-p)/Np$ and $a < b< c<k(N-p(a+1))/p(N-k) < a+1$. We suppose that $g\in L_α^{r}(\mathbb{R}^{N})$, where $r = p^*(a,c)/[p^*(a,c)-q]$ and $α= c(p^*(a,c)-q)$, is positive in a ball and that it can change sign; we also suppose that $h \in L^\infty(\mathbb{R}^k)$ and that it has a finite, positive limit $h_0$ at the origin and at infinity. To prove our results we use the Nehari manifold methods and we establish sufficient conditions to overcome the lack of compactness.

math.AP↗
arXiv2015

Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.

math.AP↗