arXiv subjects

Bhakti Bhusan Manna

Publications and source records attributed to Bhakti Bhusan Manna.

5 recordsLinked to original sources

arXiv2026

Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities

In this article, we study the semi-linear elliptic problem: \begin{equation*} -Δ_{\mathbb{B}^N} u \, - \, λu = |u|^{p-1}u, \quad u \in H^{1}(\mathbb{B}^N), \end{equation*} where $N \geq 3$, $λ> \frac{N(N-2)}{4}$, and $1 < p \leq 2^*-1$. Here, $\mathbb{B}^N$ represents the Poincaré ball model of the hyperbolic space, and $H^{1}(\mathbb{B}^N)$ denotes the Sobolev space on $\mathbb{B}^N$. In the critical case $p = 2^*-1$, with $λ< \frac{(N-1)^2}{4}$ and $N \geq 4$, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: ``the existence of sign-changing solutions in the lower dimensions $4 \leq N \leq 6$". We also prove a partial non-existence result for a large class of symmetric solutions when $λ> \frac{(N-1)^2}{4}$, based on a quantitative orbit packing condition.

math.AP↗

arXiv2026

A scalar field equation on hyperbolic space with indefinite sign nonlinearity

In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-Δ_{\mathbb{B}^N} u - λu = |u|^{p-1}u - |u|^{q-1}u, \quad u \in H^1(\mathbb{B}^N),$$ on the hyperbolic space $\mathbb{B}^N$ for $N \ge 3$. For parameters $1 < p \le 2^*-1$ (though we occasionally allow for supercritical exponents) and $q > 0$, we seek to identify the optimal spectral regimes for $λ\in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p < q$, $0 < q < 1 < p$, and $1 < q < p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p < q$, but depends solely on $N$ in the remaining cases.

math.AP↗

arXiv2025

Spike layered solutions for elliptic systems on Riemannian Manifolds

In this article, we study the following Hamiltonian system: \begin{equation*} \begin{cases} \begin{aligned} &-\varepsilon^{2}Δ_{g} u +u = |v|^{q-1}v, &-\varepsilon^{2}Δ_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M}, & \quad u,v >0 && \text{ in } \mathcal{M}, \end{aligned} \end{cases} \end{equation*} where $\mathcal{M}$ is a smooth, compact and connected Riemannian manifold of dimension $N\geq 3$ without boundary. The exponents $p,q>1$ are assumed to lie below the critical hyperbola, ensuring subcritical growth conditions. We investigate a sequence of least energy critical points of the associated dual functional and analyze their concentration behavior as $\varepsilon \to 0$. Our main result shows that the sequence of solutions exhibits point concentration, with the concentration occurring at a point where the scalar curvature of $\mathcal{M}$ attains its maximum.

math.AP↗

arXiv2025

Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -Δu - μ\frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h \times \mathbb{R}^k = \mathbb{R}^n$, with $n \geq 5$, $2 < k <n$, and $t = n - \frac{(n-2)q}{2}$. We establish the existence of a bi-radial sign-changing solution under the assumptions $0 \leq μ< \frac{(k-2)^2}{4}, \, 2 < q <2^* = \frac{2(n-k+1)}{n-k-1}$. We approach the problem by lifting it to the hyperbolic setting, leading to the equation: $-Δ_{\mathbb{B}^N} u \, - \, λu = |u|^{p-1}u, \; u \in H^1(\mathbb{B}^N)$, $\mathbb{B}^N$ is the hyperbolic ball model. We study the existence of a sign-changing solution with suitable symmetry by constructing an appropriate invariant subspace of $H^1(\mathbb{B}^N)$ and applying the concentration compactness principle, and the corresponding solution of the Hardy-Sobolev-Maz'ya type equation becomes bi-radial under the corresponding isometry.

math.AP↗

arXiv2020

Concentration results for solutions of a singularly perturbed elliptic system with variable coefficients

In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -ε^2Δu +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -ε^2Δv +c(x)v=a(x) |u|^{p-1}u &&\text{in } Ω\newline u>0, \ v>0 \text{ in } Ω, &\text{ and }\quad\frac{\partial u}{\partialν} = 0 = \frac{\partial v}{\partialν} &&\text{on }\partialΩ \end{aligned} \right. \end{equation} where $Ω$ is a smooth bounded domain in $R^n, n\geq 3$. The coefficients $a(x), b(x)$ and $c(x)$ are positive bounded smooth functions. We shall study the existence of point concentrating solutions and discuss the role of the coefficients to determine the concentration profile of the solutions. We have also discussed some applications of our main theorem towards the existence of solutions concentrating on higher-dimensional orbits.

math.AP↗