arXiv subjects

Uriel Kaufmann

Publications and source records attributed to Uriel Kaufmann.

At least 19 recordsLinked to original sources

arXiv2025

A local and nonlocal coupling model involving the $p$-Laplacian

In this paper we extend some results presented in \cite{julio} to the case of the $p$-Laplacian operator. More precisely, we consider a model that couples a local $p$-Laplacian operator with a nonlocal $p$-Laplacian operator through source terms in the equation. The resulting problem is associated with an energy functional. We establish the existence and uniqueness of a solution, which is obtained via the direct minimization of the corresponding energy functional.

math.AP↗

arXiv2024

Positive solutions for concave-convex type problems for the one-dimensional $ϕ$-Laplacian

Let $Ω=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(Ω)$, $λ,μ>0$ be real parameters, and $ϕ:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[ \begin{cases} -ϕ\left( u^{\prime}\right) ^{\prime}=λm(x)f(u)+μn(x)g(u) & \text{ in }Ω,\\ u=0 & \text{ on }\partialΩ, \end{cases} \] where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $ϕ$, respectively. Our assumptions on $ϕ$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\uı\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.

math.CA↗

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

The $\infty$-eigenvalue problem with a sign-changing weight

Let $Ω\subset\mathbb{R}^{n}$ be a smooth bounded domain and $m\in C(\overlineΩ)$ be a sign-changing weight function. For $1<p<\infty$, consider the eigenvalue problem $$ \left\{ \begin{array} [c]{ll} -Δ_{p}u=λm(x)|u|^{p-2}u & \text{in }Ω,\\ u=0 & \text{on }\partialΩ, \end{array} \right. $$ where $Δ_{p}u$ is the usual $p$-Laplacian. Our purpose in this article is to study the limit as $p\rightarrow\infty$ for the eigenvalues $λ_{k,p}\left( m\right) $ of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when $k=1$.

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↗

arXiv2017

Positive solutions for nonlinear problems involving the one-dimensional ϕ-Laplacian

Let $Ω:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( Ω\right) $ and $λ>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-ϕ\left( u^{\prime}\right) ^{\prime}+r\left( x\right) ϕ\left( u\right) $, where $ϕ:\mathbb{R\rightarrow R}$ is an odd increasing homeomorphism and $0\leq r\in L^{1}\left( Ω\right) $. We study the existence of positive solutions for problems of the form $\mathcal{L}u=λm\left( x\right) f\left( u\right)$ in $Ω,$ $u=0$ on $\partialΩ$, where $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ is a continuos function which is, roughly speaking, sublinear with respect to $ϕ$. Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases $r\equiv0$ and/or $m\geq0$.

math.CA↗

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

Positive solutions of indefinite semipositone problems via sub-super solutions

Let $Ω\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:Ω\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-Δu=λm(x)(f(u)-k)$ in $Ω$, $u=0$ on $\partialΩ$, where $λ,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $λ$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.

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↗

arXiv2015

One-dimensional singular problems involving the p-Laplacian and nonlinearities indefinite in sign

Let $Ω$ be a bounded open interval, let $p>1$ and $γ>0$, and let $m:Ω\rightarrow\mathbb{R}$ be a function that may change sign in $Ω$. In this article we study the existence and nonexistence of positive solutions for one-dimensional singular problems of the form $-(\left\vert u^{\prime}\right\vert ^{p-2}u^{\prime})^{\prime}=m\left( x\right) u^{-γ}$ in $Ω$, $u=0$ on $\partialΩ$. As a consequence we also derive existence results for other related nonlinearities.

math.CA↗

arXiv2015

On Dirichlet problems with singular nonlinearity of indefinite sign

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq1$, let $K$, $M$ be two nonnegative functions and let $α,γ>0$. We study existence and nonexistence of positive solutions for singular problems of the form $-Δu=K\left( x\right) u^{-α}-λM\left( x\right) u^{-γ}$ in $Ω$, $u=0$ on $\partialΩ$, where $λ>0$ is a real parameter. We mention that as a particular case our results apply to problems of the form $-Δu=m\left( x\right) u^{-γ}$ in $Ω$, $u=0$ on $\partialΩ$, where $m$ is allowed to change sign in $Ω$.

math.AP↗

arXiv2014

Strictly positive solutions for one-dimensional nonlinear elliptic problems

We study existence and nonexistence of strictly positive solutions for the elliptic problems of the form $Lu=m\left( x\right) u^{p}$ in a bounded open interval, with zero boundary conditions, where $L$ is a strongly uniformly elliptic differential operator, $p\in\left( 0,1\right) $, and $m$ is a function that changes sign. We also characterize the set of values $p$ for which the problem admits a solution, and in addition an existence result for other nonlinearities is presented.

math.CA↗

arXiv2013

Existence of strictly positive solutions for sublinear elliptic problems in bounded domains

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $Ω$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}ξ^{p}\leq f\left(ξ\right) \leq k_{2}ξ^{p}$ for all $ξ\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-Δu=m\left(x\right) f\left(u\right) $ in $Ω$, $u=0$ on $\partialΩ$.

math.AP↗