arXiv subjects

Kenichiro Umezu

Publications and source records attributed to Kenichiro Umezu.

18 recordsLinked to original sources

arXiv2025

Boundary layer profiles of positive solutions for logistic equations with sublinear nonlinearity on the boundary

In this paper, we consider the logistic elliptic equation $-Δu = u- u^{p}$ in a smooth bounded domain $Ω\subset \mathbb{R}^{N}$, $N\geq2$, equipped with the sublinear Neumann boundary condition $\frac{\partial u}{\partial ν} = μu^{q}$ on $\partial Ω$, where $0<q<1<p$, and $μ\geq0$ is a parameter. With sub- and super-solutions and a comparison principle for the equation, we analyze the asymptotic profile of a unique positive solution for the equation as $μ\to \infty$.

math.AP↗

arXiv2024

Diffusive logistic equation with a non Lipschitz nonlinear boundary condition arising from coastal fishery harvesting: the resonant case

For bifurcation analysis, we study the positive solution set for a semilinear elliptic equation of the logistic type, equipped with a sublinear boundary condition modeling coastal fishery harvesting. This work is a continuation of the author's previous studies, where certain results were obtained in a non resonant case, including the existence, uniqueness, multiplicity, and strong positivity for positive solutions. In this paper, we consider the delicate resonant case and develop a sort of non standard bifurcation technique at zero to evaluate the positive solution set depending on a parameter. The nonlinear boundary condition is not right-differentiable at zero.

math.AP↗

arXiv2024

Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting II

We study the positive solutions of the logistic elliptic equation with a nonlinear Neumann boundary condition that models coastal fishery harvesting ([18]). An essential role is played by the smallest eigenvalue of the Dirichlet eigenvalue problem, with respect to which a noncritical case is studied in [32]. In this paper, we extend our analysis to the critical case and further study the noncritical case for a more precise description of the positive solution set. Our approach relies on the energy method, sub- and supersolutions, and implicit function analysis.

math.AP↗

arXiv2022

Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting

Let $0 1$ is subcritical, we prove that in the case of $λ_Ω>1$, there exist at least two positive solutions for $λ>0$ sufficiently small but no positive solutions for $λ>0$ large enough. In the case of $λ_Ω<1$, there exists at least one positive solution for every $λ>0$. Here, $λ_Ω>0$ is the smallest eigenvalue of $-Δ$ under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented.

math.AP↗

arXiv2022

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions and study the uniqueness of positive solutions that this problem possesses. Superlinear elliptic problems can be expected to have multiple positive solutions under certain situations. To our end, by conducting spectral analysis for the linearized eigenvalue problem at an unstable positive solution, we find sufficient conditions for ensuring that the implicit function theorem is applicable to the unstable positive one. An application of our results to the logistic boundary condition arising from population genetics is given.

math.AP↗

arXiv2020

Uniqueness and positivity issues in a quasilinear indefinite problem

We consider the problem $$ (P_λ)\quad -Δ_{p}u=λu^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }Ω$$ under Dirichlet or Neumann boundary conditions. Here $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ ($N\geq1$), $λ\in\mathbb{R}$, $1 0$). In particular, this problem has at most one positive solution for $λ<0$. Under some condition on $a$, the above uniqueness result fails for some values of $λ>0$ as we obtain, besides the ground state solution, a \textit{second} solution positive in $Ω_{a}^{+}$. We also provide conditions on $λ$, $a$ and $q$ such that these solutions become positive in $Ω$, and analyze the formation of dead cores for a generic solution.

math.AP↗

arXiv2020

Past and recent contributions to indefinite sublinear elliptic problems

We review the indefinite sublinear elliptic equation $-Δu=a(x)u^{q}$ in a smooth bounded domain $Ω\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes sign, in which case the strong maximum principle does not apply. As a consequence, the set of nonnegative solutions of these problems has a rich structure, featuring in particular both dead core and/or positive solutions. Overall, we are interested in sufficient and necessary conditions on $a$ and $q$ for the existence of positive solutions. We describe the main results from the past decades, and combine it with our recent contributions. The proofs are briefly sketched.

math.AP↗

arXiv2020

Uniqueness and sign properties of minimizers in a quasilinear indefinite problem

Let $1<q<p$ and $a\in C(\overlineΩ)$ be sign-changing, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_{q}$ (in $W_{0}^{1,p}(Ω)$ or $W^{1,p}(Ω)$). In addition, we prove that $U_{q}$ is the only possible \textit{positive} solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_{q}$ is positive, which also guarantees that minimizers of $I_{q}$ do not change sign. Several of these results are new even for $p=2$.

math.AP↗

arXiv2020

Nonnegative solutions of an indefinite sublinear Robin problem II: local and global exactness results

We go further in the investigation of the Robin problem $(P_α)$: $-Δu=a(x)u^{q}$ in $Ω$, $u\geq0$ in $Ω$, $\partial_νu=αu$ on $\partial Ω$; on a bounded domain $Ω\subset\mathbb{R}^{N}$, with $a$ sign-changing and $0 0$. Moreover, strengthening the assumptions on $a$ and $q$ we provide a global (i.e. for every $α>0$) exactness result on the number of solutions of $(P_α)$ . Our approach also applies to the problem $(S_α)$: $-Δu=αu + a(x)u^{q}$ in $Ω$, $u\geq0$ in $Ω$, $\partial_νu=0$ on $\partial Ω$.

math.AP↗

arXiv2019

Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum

Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a smooth bounded domain, $a\in C(\barΩ)$ a sign-changing function, and $0\leq q<1$. We investigate the Robin problem \[ \begin{cases} -Δu=a(x)u^{q} & \mbox{in $Ω$},\\ u\geq0 & \mbox{in $Ω$},\\ \partial_νu=αu & \mbox{on $\partial Ω$}, \end{cases} \] where $α\in\lbrack-\infty,\infty)$ and $ν$ is the unit outward normal to $\partialΩ$. Due to the lack of strong maximum principle structure, this problem may have \textit{dead core} solutions. However, for a large class of weights $a$ we recover a \textit{positivity} property when $q$ is close to $1$, which considerably simplifies the structure of the solution set. Such property, combined with a bifurcation analysis and a suitable change of variables, enables us to show the following exactness result for these values of $q$: $(P_α)$ has \textit{exactly} one nontrivial solution for $α\leq0$, \textit{exactly} two nontrivial solutions for $α>0$ small, and \textit{no} such solution for $α>0$ large. Assuming some further conditions on $a$, we show that these solutions lie on a subcontinuum. These results rely partially on (and extend) our previous work \cite{KRQU16}, where the cases $α=-\infty$ (Dirichlet) and $α=0$ (Neumann) have been considered. We also obtain some results for arbitrary $q\in\left[ 0,1\right) $. Our approach combines mainly bifurcation techniques, the sub-supersolutions method, and \textit{a priori} lower and upper bounds.

math.AP↗

arXiv2019

A curve of positive solutions for an indefinite sublinear Dirichlet problem

We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$, of positive solutions for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$ and $a:Ω\rightarrow\mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as $q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases $u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.

math.AP↗

arXiv2018

Loop type subcontinua of positive solutions for indefinite concave-convex problems

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a non-regular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn's topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved in [15], and extend previous results established in the powerlike case.

math.AP↗

arXiv2018

A loop type component in the non-negative solutions set of an indefinite elliptic problem

We prove the existence of a loop type component of non-negative solutions for an indefinite elliptic equation with homogeneous Neumann boundary conditions. This result complements our previous results obtained in [12], where the existence of another loop type component was established in a different situation. Our proof combines local and global bifurcation theory, rescaling and regularizing arguments, a priori bounds, and Whyburn's topological method. A further investigation of the loop type component established in [12] is also provided.

math.AP↗

arXiv2017

Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:Ω\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_Ωa<0$. We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $\frac{\partial u}{\partialν}=0$ on $\partialΩ$, if $q_{0} 0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $Ω$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ ensuring the existence of a positive solution of $(P_{a,q})$. We also obtain some properties of the set of $q$'s such that $(P_{a,q})$ admits a solution which is positive on $\overlineΩ$. Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method. Several methods and results apply as well to the Dirichlet counterpart of $(P_{a,q})$.

math.AP↗

arXiv2017

An indefinite concave-convex equation under a Neumann boundary condition II

We proceed with the investigation of the problem $(P_λ): $ $-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } Ω, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial Ω$, where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_λ)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method.

math.AP↗

arXiv2016

Positivity results for indefinite sublinear elliptic problems via a continuity argument

We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some positivity results for indefinite concave-convex type problems.

math.AP↗

arXiv2016

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem $$-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } Ω, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial Ω, \leqno{(P_λ)} $$ where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|λ|$ small. We characterize then the asymptotic profiles of these solutions as $λ\to 0$, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of $(P_λ)$.

math.AP↗

arXiv2015

Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity, stability and asymptotic profiles

We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter $λ$, which is inversely proportional to the diffusion rate. We establish several multiplicity results when the diffusion rate is large and analyze the asymptotic profiles and the stability properties of these steady states as the diffusion rate grows to infinity. In particular, our results show that in some cases bifurcation from zero and from infinity occur at $λ=0$. Our approach combines variational and bifurcation techniques.

math.AP↗