arXiv subjects

Mieko Tanaka

Publications and source records attributed to Mieko Tanaka.

10 recordsLinked to original sources

arXiv2025

On Rayleigh quotients connected to $p$-Laplace equations with polynomial nonlinearities

Let $Ω$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(Ω)$-solutions of the equation $-Δ_p u = μ|u|^{q-2}u + |u|^{r-2}u$ parameterized by $μ$ are in bijection with properly normalized critical points of the $0$-homogeneous Rayleigh type quotient $R_α(u)=\|\nabla u\|_p^p/ (\|u\|_q^{αp} \|u\|_r^{p-αp})$ parameterized by $α$. We study this bijection and properties of $R_α$ for various relations between $p,q,r$. In particular, for the generalized convex-concave problem (the case $q<p<r$) the bijection allows to provide the existence and characterization of all degenerate solutions corresponding to the inflection point of the fibred energy functional: they are critical points of $R_α$ exclusively with $α= (r-p)/(r-q)$. In the subhomogeneous case $q<r \leq p$ and under additional assumptions on $Ω$, the ground state level of $R_α$ is simple and isolated, and minimizers of $R_α$ exhaust the whole set of sign-constant solutions of the corresponding equation. In the superhomogeneous case $p < q<r$, there are no sign-changing critical points in a vicinity of the ground state level of $R_α$.

math.AP↗

arXiv2023

Abstract multiplicity results for $(p,q)$-Laplace equations with two parameters

We investigate the existence and multiplicity of abstract weak solutions of the equation $-Δ_p u -Δ_q u=α|u|^{p-2}u + β|u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and $α,β\in \mathbb{R}$. We determine three generally different ranges of parameters $α$ and $β$ for which the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy. As auxiliary results, which are also of independent interest, we provide alternative characterizations of variational eigenvalues of the $q$-Laplacian using narrower and larger constraint sets than in the standard minimax definition.

math.AP↗

arXiv2022

On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-Δ_p u = λm(x)|u|^{p-2}u + ηa(x)|u|^{q-2}u + f(x)$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $q 1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian provided $m,f \in L^γ(Ω)$ with $γ>N$. For completeness, we also investigate the existence of solutions.

math.AP↗

arXiv2021

On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-Δ_p u = λ|u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $1 0\}$, when the parameter $λ$ lies in a neighborhood of the critical value $λ^* = \inf\left\{\int_Ω|\nabla u|^p \, dx/\int_Ω|u|^p \, dx: u\in W_0^{1,p}(Ω) \setminus \{0\},\ \int_Ωa|u|^q\,dx \geq 0\,\right\}$. Among main results, we show that if $p>2q$ and either $\int_Ωaφ_p^q\,dx=0$ or $\int_Ωaφ_p^q\,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $λ^*$. Here $φ_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in $Ω$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q 2q$ and $\int_Ωaφ_p^q\,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $λ^*$, two of which are strictly positive in $\{x\in Ω: a(x)>0\}$.

math.AP↗

arXiv2021

Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters

We study the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = α|u|^{p-2}u+β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(α,β)$-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(α,β) = \left(\|\nabla φ_p\|_p^p/\|φ_p\|_p^p, \|\nabla φ_p\|_q^q/\|φ_p\|_q^q\right)$, where $φ_p$ is the first eigenfunction of the $p$-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents $p$ and $q$.

math.AP↗

arXiv2021

Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = f_μ(x,u,\nabla u)$ in a bounded domain $Ω\subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_μ(x,u,\nabla u) = λ_1(p) |u|^{p-2} u + μ|u|^{q-2} u$, where $λ_1(p)$ is the first eigenvalue of the $p$-Laplacian.

math.AP↗

arXiv2019

On the Fredholm-type theorems and sign properties of solutions for $(p,q)$-Laplace equations with two parameters

We consider the Dirichlet problem for the nonhomogeneous equation $-Δ_p u -Δ_q u = α|u|^{p-2}u + β|u|^{q-2}u + f(x)$ in a bounded domain, where $p \neq q$, and $α, β\in \mathbb{R}$ are parameters. We explore assumptions on $α$ and $β$ that guarantee the resolvability of the considered problem. Moreover, we introduce several curves on the $(α,β)$-plane allocating sets of parameters for which the problem has or does not have positive or sign-changing solutions, provided $f$ is of a constant sign.

math.AP↗

arXiv2017

Remarks on minimizers for $(p,q)$-Laplace equations with two parameters

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p,q)$-Laplace equation $-Δ_p u -Δ_q u = α|u|^{p-2}u + β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $α, β\in \mathbb{R}$. A curve on the $(α,β)$-plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the $p$- and $q$-Laplacians under zero Dirichlet boundary condition are linearly independent.

math.AP↗

arXiv2017

On sign-changing solutions for $(p,q)$-Laplace equations with two parameters

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for two-parametric family of partially homogeneous $(p,q)$-Laplace equations $-Δ_p u -Δ_q u=α|u|^{p-2}u+β|u|^{q-2}u$ where $p \neq q$. By virtue of the Nehari manifolds, linking theorem, and descending flow, we explicitly characterize subsets of $(α,β)$-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the $p$- and $q$-Laplacians in one dimension.

math.AP↗

arXiv2014

On positive solutions for $(p,q)$-Laplace equations with two parameters

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-Δ_p u -Δ_q u = α|u|^{p-2} u + β|u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $Ω$. The main result of our research is the construction of a continuous curve in $(α,β)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $Ω$ for which the corresponding first Dirichlet eigenvalue of $-Δ_p$ is not monotone w.r.t. $p > 1$.

math.AP↗