arXiv subjects

Nirjan Biswas

Publications and source records attributed to Nirjan Biswas.

22 records · Page 2Linked to original sources

arXiv2022

On the optimization of the first weighted eigenvalue

For $N\geq 2$, a bounded smooth domain $Ω$ in $\mathbb{R}^N$, and $g_0, V_0 \in L^1_{loc}(Ω)$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \begin{align*} -Δ_p ϕ+ V |ϕ|^{p-2}ϕ= λg |ϕ|^{p-2}ϕ\text{ in } Ω, \quad ϕ=0 \text{ on } \partial Ω, \end{align*} where $g$ and $V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We prove the existence of a minimizing pair $(\underline{g},\underline{V})$ and a maximizing pair $(\overline{g},\overline{V})$ for $g_0$ and $V_0$ lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case $p=2$. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry.

math.AP↗

arXiv2022

On the fourth order semipositone problem in $\mathbb{R}^N$

For $N \geq 5$ and $a>0$, we consider the following semipositone problem \begin{align*} Δ^2 u= g(x)f_a(u) \text { in } \mathbb{R}^N, \, \text{ and } \, u \in \mathcal{D}^{2,2}(\mathbb{R}^N),\ \ \ \qquad \quad \mathrm{(SP)} \end{align*} where $g \in L^1_{loc}(\mathbb{R}^N)$ is an indefinite weight function, $f_a:\mathbb{R} \to \mathbb{R}$ is a continuous function that satisfies $f_a(t)=-a$ for $t \in \mathbb{R}^-$, and $\mathcal{D}^{2,2}(\mathbb{R}^N)$ is the completion of $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ with respect to $(\int_{\mathbb{R}^N} (Δu)^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of $a_1>0$ such that for each $a \in (0,a_1)$, (SP) admits a mountain pass solution. Further, we show that the mountain pass solution is positive if $a$ is near zero. For the positivity, we derive uniform regularity estimates of the solutions of (SP) for certain ranges in $(0,a_1)$, relying on the Riesz potential of the biharmonic operator.

math.AP↗

arXiv2021

Neumann eigenvalue problems on the exterior domains

For $ p\in (1, \infty)$, we consider the following weighted Neumann eigenvalue problem on $B_1^c$, the exterior of the closed unit ball in $R^N$: \begin{equation}\label{Neumann eqn} \begin{aligned} -Δ_p ϕ& = λg |ϕ|^{p-2} ϕ\ \text{in}\ B^c_1, \\ \displaystyle\frac{\partial ϕ}{\partial ν} &= 0 \ \text{on} \ \partial B_1, \end{aligned} \end{equation} where $Δ_p$ is the $p$-Laplace operator and $g \in L^1_{loc}(B^c_1)$ is an indefinite weight function. Depending on the values of $p$ and the dimension $N$, we take $g$ in certain Lorentz spaces or weighted Lebesgue spaces and show that the above eigenvalue problem admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of $W^{1,p}(B^c_1)$ into $L^p(B^c_1, |g|)$ for $g$ in certain weighted Lebesgue spaces. For $N>p$, we also provide an alternate proof for the embedding of $W^{1,p}(B^c_1)$ into $L^{p^*,p}(B^c_1)$. Further, we show that the set of all eigenvalues is closed.

math.AP↗

arXiv2021

On global bifurcation for the nonlinear Steklov problems

For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $Ω$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -Δ_p ϕ& = 0 \; \text{in} \ Ω, \\ |\nabla ϕ|^{p-2} \frac{\partial ϕ}{\partial ν} &= λ\left( g |ϕ|^{p-2}ϕ+ f r(ϕ) \right) \; \text{on} \ \partial Ω, \end{aligned} \end{equation*} where $Δ_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial Ω)$ are indefinite weight functions and $r \in C(\mathbb R)$ satisfies $r(0)=0$ and certain growth conditions near zero and at infinity. For $f,g$ in some appropriate Lorentz-Zygmund spaces, we establish the existence of a continuum that bifurcates from $(λ_1,0)$, where $λ_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem \begin{equation*} \begin{aligned} -Δ_p ϕ& = 0 \; \text{in} \ Ω, \\ |\nabla ϕ|^{p-2} \frac{\partial ϕ}{\partial ν} &= λg |ϕ|^{p-2}ϕ\ \text{on} \ \partial Ω. \end{aligned} \end{equation*}

math.AP↗