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Nirjan Biswas

Publications and source records attributed to Nirjan Biswas.

At least 19 recordsLinked to original sources
arXiv2026

Critical fractional $p$-Hardy Sobolev equations: Global compactness and multiplicity of positive solutions

We study the critical fractional $p$-Hardy-Sobolev equation \begin{equation}\tag{$\mathcal{P}$}\label{a-main} (-Δ_p)^s u -μ\dfrac{|u|^{p-2}u}{|x|^{sp}}=\dfrac{|u|^{p^*_s(α)-2}u}{|x|^α}+f \;\mbox{ in }\,\mathbb{R}^d, \quad u\in \mathcal{D}^{s,p}(\mathbb{R}^d), \end{equation} where $1 0$, $p^*_s(α):= p(d-α)/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $(\mathcal{D}^{s,p}(\mathbb{R}^d))^*$. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When $α>0$, the loss of compactness is described by dilations of solutions of the Hardy-Sobolev limit problem. The case $α=0$ has a different structure: in addition to Hardy profiles, pure Sobolev profiles may occur when the centre of concentration escapes from the Hardy singularity relative to its scale. We give a direct centre-scale analysis of these two concentration regimes and obtain the corresponding energy decomposition and profile separation. As an application, under an explicit smallness assumption on $f$, we first obtain a positive solution for \eqref{a-main} with negative energy. We then construct a nonlinear path based on hidden convexity whose energy remains strictly below the first bubbling threshold. A minimax argument, combined with the global compactness theorem, then yields a second distinct positive solution for \eqref{a-main}.

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arXiv2026

On $p$-fractional weakly-coupled system with critical nonlinearities

This paper deals with the following nonlocal system of equations: \begin{equation}\tag{$\mathcal S$}\label{MAT1} (-Δ_p)^s u = \fracα{p_s^*}|u|^{α-2}u|v|^β+f(x) \text{ in } \mathbb{R}^{d}, \, (-Δ_p)^s v = \fracβ{p_s^*}|v|^{β-2}v|u|^α+g(x) \text{ in } \mathbb{R}^{d},\; u,v >0 \mbox{ in } \mathbb{R}^{d}, \end{equation} where $0 sp$, $α,β>1$, $α+β=\frac{dp}{d-sp}$, and $f,g$ are nontrivial nonnegative functionals in the dual space of $\mathcal{D}^{s,p}(\mathbb{R}^{d})$. The primary objective of this paper is to present a global compactness result that offers a complete characterization of the Palais-Smale sequences of the energy functional associated with \eqref{MAT1}. Using this characterization, within a certain range of $s$, we establish the existence of a solution with negative energy for \eqref{MAT1} when $\ker(f)=\ker(g)$, and $\|f\|_{(\mathcal{D}^{s,p})'}, \|g\|_{(\mathcal{D}^{s,p})'}$ are small.

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arXiv2026

Regularity for the fractional logarithmic $p$-Laplacian

We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-Δ_p)^{s+\log} u:= \frac{\rm d}{{\rm d}t}(-Δ_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.

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arXiv2026

Brezis-Nirenberg problems for mixed local-nonlocal operators with superlinear perturbations: compactness and applications

In this paper, we consider the following mixed local nonlocal Brezis-Nirenberg problem \begin{equation}\label{crit_pro_abstract}\tag{$\mathcal{P}_{2^*}$} -Δu+(-Δ)^s u=λ|u|^{p-2}u+|u|^{2^*-2}u \text{ in } Ω,\quad u=0 \text{ in } \mathbb{R}^N \setminus Ω, \end{equation} where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $N\geq3$, $s\in(0,1)$, $λ>0$, and $2\leq p<2^*=\frac{2N}{N-2}$. We establish a compactness result for the following class of subcritical/critical problems \begin{equation}\label{sub_pro_abstract}\tag{$\mathcal{P}_{p_n}$} -Δu+(-Δ)^s u=λ|u|^{p-2}u+|u|^{p_n-2}u \text{ in } Ω,\quad u=0 \text{ in } \mathbb{R}^N \setminus Ω, \end{equation} where $p_n \in (p,2^* ]$ and $p_n\to 2^*$. Specifically, for $p \in (2+\frac{4s}{N-2},2^*)$ when $N>6-4s$, and for $p \in (2^*-1,2^*)$ when $N\leq6-4s$, we prove that any bounded sequence of solutions $\{u_n\}$ to \eqref{sub_pro_abstract} is relatively compact in the energy space, and converges strongly to a nontrivial solution to \eqref{crit_pro_abstract}. To the best of our knowledge, this is the first paper to address this type of compactness result for a non-homogeneous operator. Due to the presence of the non-homogeneous operator, proving the compactness result requires several delicate new and novel estimates, which we believe will be of independent interest for further studies of related problems. As an application of this compactness result, under the same ranges of $N$ and $p$, we prove that \eqref{crit_pro_abstract} admits infinitely many sign-changing solutions.

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arXiv2026

Harnack inequality for mixed local-nonlocal weighted homogeneous equations

We consider the following class of mixed local-nonlocal equations: \begin{equation}\tag{P}\label{eq:P} -Δ_p u + (-Δ)_p^s u = V |u|^{p-2}u \text{ in } Ω, \end{equation} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(Ω)$ where $q>d/p$ when $d>p$ and $q>1$ when $d \le p$. We establish the Harnack inequality for a weak solution and the weak Harnack inequality for a weak supersolution to \eqref{eq:P}. Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by $(-Δ)_p^s$. This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions.

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arXiv2026

Mixed local-nonlocal equations with critical nonlinearity on $\mathbb{R}^N$: Non-existence, Existence, and Multiplicity of positive solutions

We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract_1}\tag{$\mathcal{P}_p$} -Δ_p u+(-Δ_p)^s u=|u|^{p^*-2}u+f(x)\text{ in }\mathbb{R}^N, \end{equation} where $s \in (0,1), p \in (1, \infty), N>p$, $p^*=\frac{Np}{N-p}$, and $f$ is a nonnegative functional in the dual space of the ambient solution space. If $f \equiv0$, then we show that \eqref{main_prob_abstract_1} does not admit any nontrivial weak solution. This phenomenon stands in contrast to the purely local and purely nonlocal cases. On the other hand, if $f$ is a nontrivial nonnegative functional, we establish the existence of a positive weak solution to \eqref{main_prob_abstract_1} provided $\|f\|$ is small. For this purpose we prove the concentration compactness principle for the mixed operator $-Δ_p +(-Δ_p)^s$ in $\mathbb{R}^N$. We also discuss the multiplicity of positive weak solutions to \eqref{main_prob_abstract_1}.

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arXiv2026

Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions

We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian, \begin{equation}\tag{$\mathcal{P}_{λ,\varepsilon}$} -Δ_p u+\varepsilon(-Δ_p)^s u=λ|u|^{q-2}u+|u|^{p^*-2}u \;\text{ in }Ω,\quad u=0 \; \text{ in }\mathbb{R}^N \setminus Ω, \end{equation} where $Ω\subset\mathbb{R}^N$ is a bounded open set, $ε\in(0,1]$, $0 0$, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of $Λ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has a positive minimal solution for $0<λ<Λ_\varepsilon$, a positive solution for $λ=Λ_\varepsilon$ and no positive solution for $λ>Λ_\varepsilon$. We also prove the existence of $0<λ^\#\leqΛ_\varepsilon$ such that ($\mathcal{P}_{λ,\varepsilon}$) has at least two positive solutions for $λ\in(0,λ^\#)$ provided $\varepsilon$ small enough. This extends the recent result of Biagi and Vecchi (Nonlinear Anal. 256 (2025),113795), Amundsen, et al. (Commun. Pure Appl. Anal., 22(10):3139-3164, 2023) from $p=2$ to the general $1<p<N$. Additionally, it extends the classical result of Azorero and Peral (Indiana Univ. Math. J., 43(3):947-957, 1994) to the mixed local-nonlocal quasilinear problems. Moreover, our results complements the multiplicity results for nonnegative solutions in da Silva, et al. (J. Differential Equations, 408:494-536, 2024).

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arXiv2026

Fractional $p$-Laplace systems with critical Hardy nonlinearities: Existence and Multiplicity

Let $Ω\subset \mathbb{R}^d$ be a bounded open set containing zero, $s \in (0,1)$ and $p \in (1, \infty)$. In this paper, we first deal with the existence, non-existence and some properties of ground-state solutions for the following class of fractional $p$-Laplace systems \begin{equation*} \left\{\begin{aligned} &(-Δ_p)^s u= \fracα{q} \frac{|u|^{α-2}u|v|^β}{|x|^m} \;\;\text{in}\;Ω,\\ &(-Δ_p)^s v= \fracβ{q} \frac{|v|^{β-2}v|u|^α}{|x|^m}\;\;\text{in}\;Ω,\\ &u=v=0\, \mbox{ in }\mathbb{R}^d\setminus Ω, \end{aligned} \right. \end{equation*} where $d>sp$, $α+ β= q$ where $p \leq q \leq p_{s}^{*}(m)$ where $p_{s}^{*}(m) = \frac{p(d-m)}{d-sp}$ with $0 \leq m \le sp$. Additionally, we establish a concentration-compactness principle related to this homogeneous system of equations. Next, the main objective of this paper is to study the following non-homogenous system of equations \begin{equation*} \left\{\begin{aligned} &(-Δ_p)^s u = η|u|^{r-2}u + γ\fracα{p_{s}^{*}(m)} \frac{|u|^{α-2}u|v|^β}{|x|^m} \;\;\text{in}\;Ω,\\ &(-Δ_p)^s v = η|v|^{r-2}v + γ\fracβ{p^{*}_{s}(m)} \frac{|v|^{β-2}v|u|^α}{|x|^m}\;\;\text{in}\;Ω,\\ &u=v=0\, \mbox{ in }\mathbb{R}^d\setminus Ω, \end{aligned} \right. \end{equation*} where $η, γ> 0$ are parameters and $p \leq r < p_{s}^{*}(0)$. Depending on the values of $η, γ$, we obtain the existence of a non semi-trivial solution with the least energy. Further, for $m=0$, we establish that the above problem admits at least $\text{cat}_Ω(Ω)$ nontrivial solutions.

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arXiv2025

Lipschitz potential estimates for diffusion with jumps

For $p \in (1, \infty)$ and $s \in (0,1)$, we consider the following mixed local-nonlocal equation $$ - Δ_p u + (-Δ_p)^s u = f \; \text{in} \; Ω,$$ where $Ω\subset \mathbb{R}^d$ is a bounded domain and the function $f \in L_{loc}^1(Ω)$. Depending on the dimension $d$, we prove gradient potential estimates of weak solutions for the entire ranges of $p$ and $s$. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $p=2$.

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arXiv2025

On semipositone problems over $\mathbb{R}^N$ for the fractional $p$-Laplace operator

For $N \geq 1, s\in (0,1)$, and $p \in (1, \frac{N}{s})$ we find a positive solution to the following class of semipositone problems associated with the fractional $p$-Laplace operator: \begin{equation}\tag{SP} (-Δ)_{p}^{s}u = g(x)f_a(u) \text{ in } \mathbb{R}^N, \end{equation} where $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is a positive function, $a>0$ is a parameter and $f_a \in \mathcal{C}(\mathbb{R})$ is defined as $f_a(t) = f(t)-a$ for $t \ge 0$, $f_a(t) = -a(t+1)$ for $t \in [-1, 0]$, and $f_a(t) = 0$ for $t \le -1$, where $f$ is a non-negative continuous function on $[0,\infty)$ satisfies $f(0)=0$ with subcritical and Ambrosetti-Rabinowitz type growth. Depending on the range of $a$, we obtain the existence of a mountain pass solution to (SP) in $\mathcal{D}^{s,p}(\mathbb{R}^N)$. Then, we prove mountain pass solutions are uniformly bounded with respect to $a$, over $L^r(\mathbb{R}^N)$ for every $r \in \left[\frac{Np}{N-sp}, \infty\right]$. In addition, if $p>\frac{2N}{N+2s}$, we establish that (SP) admits a non-negative mountain pass solution for each $a$ near zero. Finally, under the assumption $g(x) \leq \frac{B}{|x|^{β(p-1)+sp}}$ for $B>0, x \neq 0$, and $ β\in \left(\frac{N-sp}{p-1}, \frac{N}{p-1}\right)$, we derive an explicit positive radial subsolution to (SP) and show that the non-negative solution is positive a.e. in $\mathbb{R}^N$.

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arXiv2025

On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters

For $s_1,s_2\in(0,1)$ and $p,q \in (1, \infty)$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $α, β\in \mathbb{R}$ driven by the sum of two nonlocal operators: \begin{equation*} (-Δ)^{s_1}_p u+(-Δ)^{s_2}_q u=α|u|^{p-2}u+β|u|^{q-2}u\;\;\text{in }Ω, \quad u=0\;\;\text{in } \mathbb{R}^d \setminus Ω, \ \ \ \qquad \quad \mathrm{(P)} \end{equation*} where $Ω\subset \mathbb{R}^d$ is a bounded open set. Depending on the values of $α,β$, we completely describe the existence and non-existence of positive solutions to (P). We construct a continuous threshold curve in the two-dimensional $(α, β)$-plane, which separates the regions of the existence and non-existence of positive solutions. In addition, we prove that the first Dirichlet eigenfunctions of the fractional $p$-Laplace and fractional $q$-Laplace operators are linearly independent, which plays an essential role in the formation of the curve. Furthermore, we establish that every nonnegative solution of (P) is globally bounded.

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arXiv2024

Strict Faber-Krahn type inequality for the mixed local-nonlocal operator under polarization

Let $Ω\subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class $\mathcal{C}^{1,β}$ for some $β\in (0,1)$. For $p\in (1, \infty )$ and $s\in (0,1)$, let $Λ^s_{p}(Ω)$ be the first eigenvalue of the mixed local-nonlocal operator $-Δ_p+(-Δ_p)^s$ in $Ω$ with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber-Krahn type inequality for $Λ_{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of $Λ_{p}^s(\cdot )$ over annular domains and characterize the rigidity property of the balls in the classical Faber-Krahn inequality for $-Δ_p+(-Δ_p)^s$.

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arXiv2024

Higher order fractional weighted homogeneous spaces: characterization and finer embeddings

In this article, for $N \geq 2, s \in (1,2), p\in (1, \frac{N}{s}), σ=s-1 $ and $a \in [0, \frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space \begin{align*} {\mathcal D}^{s,p}_a(\mathbb{R}^N) := \overline{\mathcal{C}_c^{\infty}(\mathbb{R}^N)}^{[\cdot]_{s,p,a}} \text{ where } [u]_{s,p,a} := \left( \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{\left| \nabla u(x) -\nabla u(y) \right|^p}{\left|x-y \right|^{N+σp}} \, \frac{\mathrm{d}x}{|x|^a} \frac{\mathrm{d}y}{|y|^a} \right)^{\frac{1}{p}}, \end{align*} and higher order fractional weighted homogeneous space \begin{align*} \mathring{W}^{s,p}_a(\mathbb{R}^N):= \left\{u \in L_a^{p^*_s}(\mathbb{R}^N): \| \nabla u \|_{L_a^{p^*_σ}(\mathbb{R}^N)} + [u]_{s,p,a} < \infty \right\} \end{align*} with the weighted Lebesgue norm \begin{align*} \| u \|_{L_a^{p^*_α}(\mathbb{R}^N)}:= \left( \int_{\mathbb{R}^N} \frac{ |u(x)|^{p^*_α}}{|x|^{\frac{2ap^*_α}{p}}} \, {\mathrm{d}x} \right)^{\frac{1}{p^*_α}}, \text{ where } p^*_α=\frac{Np}{N-αp} \text{ for } α= s,σ. \end{align*} To achieve this, we prove that $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ is dense in $\mathring{W}^{s,p}_a(\mathbb{R}^N)$ with respect to $[\cdot]_{s,p,a}$, and $[\cdot]_{s,p,a}$ is an equivalent norm on $\mathring{W}^{s,p}_a(\mathbb{R}^N)$. Further, we obtain a finer embedding of ${\mathcal D}^{s,p}_a(\mathbb{R}^N)$ into the Lorentz space $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N)$, where $L^{\frac{Np}{N-sp-2a}, p}(\mathbb{R}^N) \subsetneq L_a^{p^*_s}(\mathbb{R}^N)$.

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arXiv2024

Study of fractional semipositone problems on $\mathbb{R}^N$

Let $s \in (0,1)$ and $N >2s$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-Δ)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$.

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arXiv2024

Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli

Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that $B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in (1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with $p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let $λ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-Δ_p)^s$ in $B\setminus \overline{B'}$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that $λ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for $λ_{p,q}^s(\cdot )$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of $(-Δ)^s$ and $q=1, 2$ to $(-Δ_p)^s$ and $q\in [1, p^*_s).$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in $\mathbb{R}^d$.

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arXiv2022

Admissible function spaces for weighted Sobolev inequalities

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $Ω=Ω_1 \times Ω_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type inequality: \begin{align} \int_Ω |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_Ω | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(Ω), \end{align} for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we give a sufficient condition on $g_1,g_2$ so that the best constant in the above inequality is attained in the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$-the completion of $\mathcal{C}^1_c(Ω)$ with respect to $\|\nabla u\|_{L^p(Ω)}$.

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