M. Alif
We obtain a description of the Fučik spectrum associated to the one-dimensional asymmetric problem with indefinite weights $Lu = am(t)u^+ - bn(t)u^-$ in $]T_1, T_2[$, $u'(T_1) = 0 = u'(T_2)$, where $L$ is a Sturm-Liouville operator. Our approach is based on the shooting method.
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This paper is devoted to the study of the Fu/v cik spectrum with indefinite weights in the one-dimensional case.
We prove the existence of a pair of positive radial solutions for the Neumann boundary value problem \begin{equation*} \begin{cases} \, \mathrm{div}\,\Biggl{(} \dfrac{\nabla u}{\sqrt{1- | \nabla u |^{2}}}\Biggr{)} + λa(|x|)u^p = 0, & \text{in $B$,} \\ \, \partial_νu=0, & \text{on $\partial B$,} \end{cases} \end{equation*} where $B$ is a ball centered at the origin, $a(|x|)$ is a radial sign-changing function with $\int_B a(|x|)\,\mathrm{d}x < 0$, $p>1$ and $λ> 0$ is a large parameter. The proof is based on the Leray-Schauder degree theory and extends to a larger class of nonlinearities.
We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{λ,μ}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{λ,μ}$ is a step-wise indefinite weight function, precisely $a_{λ,μ}\equivλ$ in $[0,σ]\cup[1-σ,1]$ and $a_{λ,μ}\equiv-μ$ in $(σ,1-σ)$, for some $σ\in\left(0,\frac{1}{2}\right)$, with $λ$ and $μ$ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in $(0,1)$ as $λ$ and $μ$ vary. Depending on $λ$ and based on a phase-plane analysis and on time-map…
We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for $0<s<1$. As a consequence, the best strategy for survival is either following the diffusion with $s=1$…
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.…
The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of a Neumann boundary value problem of superlinear indefinite type related to the mean curvature operator with a sublinear potential at infinity.
A uniqueness result for two-step indefinite $p$-Laplacian equations
本文证明了两步不定权 p-Laplacian 方程在 Neumann 和周期边界条件下正解的唯一性猜想，并给出所有参数范围内的完整存在性与唯一性分类，数值例子显示非两步权可产生多解。
We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain $Ω\subset \mathbb{R}^{N}$, within a suitable class of sign-changing weights. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of $D $ tends to zero. We show that, when the measure of $D$ is sufficiently small, $u $ has a unique local maximum point lying on the boundary of $Ω$ and $D$ is connected. Furthermore, …
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 $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. 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 $a$ e…
Optimization of the principal eigenvalue of the Neumann Laplacian with indefinite weight and monotonicity of minimizers in cylinders
本文研究了带不定权重的 Neumann Laplacian 主特征值优化问题，证明了极小化子的存在性与特征刻画，并揭示了柱体区域中极小化子沿母线方向的单调性。
An indefinite concave-convex equation under a Neumann boundary condition II
本文研究Neumann边界条件下含不定凹凸非线性项的方程，在$b\geq0$且非恒零时，利用先验界、正则化和Whyburn方法证明非平凡非负解的无界子连续统存在。
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 posit…
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.
Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour
本文研究不定权 Minkowski 平均曲率 Neumann 问题，利用 Szulkin 非光滑理论证明大参数下两个正解的存在性，并分析其渐近行为与几何约束饱和特性。
Loop type subcontinua of positive solutions for indefinite concave-convex problems
针对带不定权的凹凸型椭圆方程，论文结合正则化、先验界与 Whyburn 拓扑方法处理零处不可微难题，证明了 Dirichlet 和 Neumann 条件下非负解的环型子连续统及其正性。
Past and recent contributions to indefinite sublinear elliptic problems
本文综述不定次线性椭圆方程 −Δu=a(x)u^q 在齐次 Dirichlet 或 Neumann 边界下的正解存在性条件，整合历史结果与作者近期贡献，并简述相关证明。
Homogenization of an indefinite spectral problem arising in population genetics
研究渐近细圆柱中二阶自伴椭圆算子的不定谱问题，推导出谐振子形式的一维有效谱问题并证明谱收敛性，为群体遗传学中带变号权重的谱问题提供理论支持。
In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalueproblem in a fixed box $Ω$ and we consider the general case of Robin boundary conditions on $\partialΩ$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be pa…
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 $\mathbb{R}^{N}$ ($N\geq1$), $λ\in\mathbb{R}$, $1<q<p$, and $a\in C(\overlineΩ)$ changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for $λ<0$ the functional \[ I_λ(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{λ}{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] defined in $X=W_{0}^{1,p}(Ω)$ or $X=W^{1,p}(Ω)$, ha…
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…
Weighted Sobolev Spaces and an Elliptic Eigenvalue Problem with an Indefinite Weight on Domains in Noncompact Riemannian Manifolds
研究非紧黎曼流形上带不定权的椭圆特征值问题，通过建立条件确保加权Sobolev嵌入和迹算子性质，证明该问题有无限多个正负特征值，建立弱特征函数有界性及特定衰减性质。
Neumann eigenvalue problems on the exterior domains
该文研究外区域上带不定权的p-Laplace型Neumann特征值问题，通过建立对应Sobolev空间的紧嵌入结论，证明存在含唯一主特征值的无界正特征值序列，且特征值集为闭集。
We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box $Ω\subset\mathbb{R}^N$, which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is lead to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, when…
Sign-indefinite transmission conditions and near-zero eigenvalues and resonances
该研究针对超材料建模的线性算子，探讨不定号传输条件下的近零特征值与共振问题，明确算子谱特性，给出零为阈值特征值的参数条件，并分析其在参数小扰动下的分支行为。
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