arXiv subjects

Rakesh Arora

Publications and source records attributed to Rakesh Arora.

At least 19 recordsLinked to original sources

arXiv2026

On fractional $1$-Laplacian evolution equation

We study a nonlocal evolution problem driven by the fractional $1$-Laplacian with a Carath\'eodory nonlinearity satisfying subcritical growth conditions. We develop a potential well framework to investigate the existence and qualitative behavior of solutions at different initial energy levels. By combining a modified potential well method with Galerkin approximations, we establish the global existence of weak and strong solutions under appropriate conditions on the initial energy and the sign of the associated Nehari functional. To address the singular nature of the fractional $1$-Laplacian, we approximate the problem by a family of fractional $p$-Laplacian equations, derive estimates uniform with respect to $p>1$, and pass to the limit as $p\to 1^{+}$. Furthermore, in the low-dimensional regime $N<2s$, we employ a subdifferential approach to establish the local existence of strong solutions and investigate their qualitative behavior.

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arXiv2026

Global Calderon-Zygmund estimates for irregular double-phase evolution problem with non-divergence data

We study irregular double-phase parabolic equations with variable exponents and non-divergence data, \[ u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] under the homogeneous Dirichlet boundary conditions. Here, $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain, $T>0$, \[ \mathcal{F}(z,\nabla u)=a(z)|\nabla u|^{p(z)-2} + b(z) |\nabla u |^{q(z)-2} \] with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. The nonnegative coefficients $a(z), b(z)$ satisfy the inequality $a(z)+b(z)>0$ in $Q_T$, the space and time derivatives of $a$ and $b$ belong to $L^d(Q_T)$ with some $d$ depending on the data. If \[ f\in L^\sigma(Q_T) \quad \text{for} \ \sigma \in (2, N+2] \quad \text{and} \quad \mathcal{F}((\cdot,0),\nabla u_0)\,|\nabla u_0|^{r+2}\in L^1(\Omega), \] where $0\le r\le K(N,\sigma,p,q)$ if $\sigma<N+2$, while $r\ge0$ is arbitrary if $\sigma=N+2$, then the problem has a unique strong solution, for which we prove the global transfer of integrability from the initial data and the forcing term to the double-phase flux in the spirit of Calder\'on-Zygmund theory, higher integrability of the gradient, and the second-order space regularity: \[ \begin{split} & \text{$\mathcal{F}((\cdot,t),\nabla u(\cdot,t))|\nabla u(\cdot,t)|^{r+2}\in L^1(\Omega)$ for a.e. $t\in (0,T)$}, \\ & \text{$|\nabla u|^{2(\min\{p(z),q(z)\}-1)+r+s}\in L^1(Q_T)$ for every $s\in\left(0,\frac{4}{N+2}\right)$}, \\ & \mathcal{F}(z,\nabla u)|\nabla u|^{\frac{r+2}{2}} \in L^2(0,T;W^{1,2}(\Omega)). \end{split} \] The results improve and complement the results in \cite{Arora-Shmarev-JGA-2026} and extend them to the full range $r \geq 0$.

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arXiv2026

Spectral Properties of the Logarithmic Laplacian with Indefinite Weights

In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusternik-Schnirelman eigenvalues and show that the first eigenvalue is simple, with the associated eigenfunction having constant sign in the domain. In contrast, eigenfunctions corresponding to higher eigenvalues necessarily change sign. We further establish a nodal domain type inequality relating the higher eigenvalues to the measure of the positive and negative parts of the corresponding eigenfunctions, which is of independent interest. As an application, we prove that the first eigenvalue is isolated. In addition, we obtain alternative variational characterizations of the first and second eigenvalues and establish monotonicity properties of the eigenvalues with respect to both the weight function and the domain.

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arXiv2026

On the Fu\v{c}\'{i}k spectrum of the Logarithmic Laplacian

In this paper, we investigate the Fu\v{c}\'{i}k spectrum $\Sigma_L$ associated with the logarithmic Laplacian. This spectrum is defined as the set of all pairs $(\alpha,\beta) \in \mathbb{R}^2$ for which the problem \[ L_\Delta u = \alpha u^+-\beta u^- ~\text{in} ~ \Omega \quad \text{and} \quad u=0 ~\text{in} ~\mathbb{R}^N\setminus \Omega \] admits a nontrivial solution $u$. Here, $\Omega \subset \mathbb{R}^N$ is a bounded domain with $C^{1,1}$ boundary, $u^\pm = \max\{\pm u,0\}$, and $u = u^+ - u^-$. We show that the lines $\lambda_1^L \times \mathbb{R}$ and $\mathbb{R} \times \lambda_1^L$, where $\lambda_1^L$ denotes the first eigenvalue of $L_\Delta$, lies in the spectrum $\Sigma_L$ and are isolated within the spectrum. Furthermore, we establish the existence of the first nontrivial curve in $\Sigma_L$ and analyze its qualitative properties, including Lipschitz continuity, strict monotonicity, and asymptotic behavior. In addition, we obtain a variational characterization of the second eigenvalue of the logarithmic Laplacian and show that all eigenfunctions corresponding to eigenvalues $\lambda > \lambda_1^L$ are sign-changing. Finally, we address a nonresonance problem with respect to the Fu\v{c}\'{i}k spectrum $\Sigma_L$, employing variational methods and carefully overcoming the difficulties arising from the contrasting features of the first eigenvalue $\lambda_1^L$.

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arXiv2025

Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian

In this work, we show the existence of an unbounded sequence of minimax eigenvalues for the logarithmic $p$-Laplacian via the $\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz. As an application of these minimax eigenvalues and $p$-logarithmic Sobolev inequality proved in [4], we prove new existence results for nonlocal Dirichlet problems involving logarithmic $p$-Laplacian and nonlinearities with $p$-superlinear and subcritical growth at infinity.

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arXiv2025

Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth

In this paper, we derive a new $p$-Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic $p$-Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic $p$-Laplacian and critical growth nonlinearities perturbed with superlinear-subcritical growth terms. By employing the method of the Nehari manifold, we prove the existence of a nontrivial weak solution. Lastly, we conduct an asymptotic analysis of a weighted nonlocal, nonlinear problem governed by the fractional $p$-Laplacian with superlinear or sublinear type non-linearity, demonstrating the convergence of least energy solutions to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic $p$-Laplacian as the fractional parameter $s \to 0^+$. The findings in this work serve as a nonlinear analogue of the results reported in \cite{Angeles-Saldana, Arora-Giacomoni-Vaishnavi, Santamaria-Saldana}, thereby extending their scope to a broader variational framework.

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arXiv2025

Irregular double-phase evolution problem: existence and global regularity

We investigate the homogeneous Dirichlet problem for the irregular double-phase evolution equation \[ u_t-\operatorname{div} \left( a(z)|\nabla u|^{p(z)-2} \nabla u + b(z)|\nabla u|^{q(z)-2} \nabla u\right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] where $\Omega \subset \mathbb{R}^N$, $N \geq 2$ is a bounded domain, $T>0$, The non-differentiable coefficients $a(z)$, $b(z)$, the free term $f$, and the variable exponents $p$, $q$ are given functions. The coefficients $a$ and $b$ are nonnegative, bounded, satisfy the inequality \[ a(z)+b(z)\geq \alpha \quad \text{in} \ Q_T, \quad \text{and} \quad |\nabla a|, |\nabla b|, a_t, b_t \in L^d(Q_T) \] for some constant $\alpha>0$, and with $d>2$ depending on $\sup p(z)$, $\sup q(z)$, $N$, and the regularity of initial data $u(x,0)$. The free term $f$ and initial data $u(x,0)$ satisfy \[ f\in L^\sigma(Q_T) \ \text{with} \ \sigma>2 \quad \text{and} \quad |\nabla u(x,0)|\in L^{r}(\Omega) \ \text{with} \ r\geq \max \bigg\{2,\sup_{Q_T}p(z),\sup_{Q_T}q(z)\bigg\}. \] The variable exponents $p,q \in C^{0,1}(\overline{Q}_T)$ satisfy the balance condition \[ \frac{2N}{N+2} < p(z), q(z)< +\infty \ \text{in} \ \overline Q_T \quad \text{and} \quad \max\limits_{\overline Q_T}|p(z)-q(z)|< \dfrac{2}{N+2}. \] Under the above assumptions, we establish the existence of a solution, which is obtained as the limit of classical solutions to a family of regularized problems and preserves initial temporal integrability: \[ |\nabla u(\cdot, t)| \in L^r(\Omega) \ \text{for a.e.} \ t \in (0,T), \] gains global higher integrability: \[ |\nabla u|^{\min\{p(z), q(z)\} + s +r} \in L^1(Q_T) \ \text{for any} \ s \in \left(0, \frac{4}{N+2}\right), \] and attains second-order regularity: \[ a(z) |\nabla u|^{\frac{p+r-2}{2}}+b(z) |\nabla u|^{\frac{q+r-2}{2}}\in L^2(0,T;W^{1,2}(\Omega)). \]

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arXiv2025

The Brezis-Nirenberg and logistic problem for the Logarithmic Laplacian

In this work, we study the non-local analogue of Brezis-Nirenberg and logistic type elliptic equations involving the logarithmic Laplacian and critical logarithmic non-linearity with superlinear-subcritical perturbation. In the first part of this work, we derive new sharp, continuous and compact embeddings of nonlocal Sobolev spaces (of order zero) into Orlicz type spaces. As an application of these embeddings and variational analysis as carried out in \cite{Angeles-Saldana-2023, Santamaria-Saldana-2022}, we prove the existence of a least energy weak solution of the Brezis-Nirenberg and logistic type problem involving the logarithmic Laplacian. For the uniqueness of solution, we prove a new D\'iaz-Saa type inequality, which is of independent interest and can be applied to a larger class of problems. In the second part of the work, depending upon the growth of non-linearity and regularity of the weight function, we study the small-order asymptotic of non-local weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that least energy solutions of a weighted non-local fractional problem with superlinear or sublinear type non-linearity converge to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic Laplacian.

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arXiv2025

Global Existence and Finite-Time Blow-Up of Solutions for Parabolic Equations Involving the Fractional Musielak $g_{x,y}$-Laplacian

In this work, we study the parabolic fractional Musielak $g_{x,y}$-Laplacian equation: \begin{equation*} \left\{ \begin{aligned} u_{t} + (-\Delta)_{{g}_{x,y}}^{s} u &= f(x,u), && \text{in } \Omega \times (0, \infty), u &= 0, && \text{on } \mathbb{R}^N \setminus \Omega \times (0, \infty), u(x,0) &= u_0(x), && \text{in } \Omega, \end{aligned} \right. \end{equation*} where $(-\Delta)_{{g}_{x,y}}^{s}$ denotes the fractional Musielak $g_{x,y}$-Laplacian, and $f$ is a Carath\'eodory function satisfying subcritical growth conditions. Using the modified potential well method and Galerkin's method, we establish results on the local and global existence of weak and strong solutions, as well as finite-time blow-up, depending on the initial energy level (low, critical, or high). Moreover, we explore a class of nonlocal operators to highlight the broad applicability of our approach. This study contributes to the developing theory of fractional Musielak-Sobolev spaces, a field that has received limited attention in the literature. To our knowledge, this is the first work addressing the parabolic fractional $g_{x,y}$-Laplacian equation.

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arXiv2025

Logarithmic double phase problems with generalized critical growth

In this paper we study logarithmic double phase problems with variable exponents involving nonlinearities that have generalized critical growth. We first prove new continuous and compact embedding results in order to guarantee the well-definedness by studying the Sobolev conjugate function of our generalized $N$-function. In the second part we prove the concentration compactness principle for Musielak-Orlicz Sobolev spaces having logarithmic double phase modular function structure. Based on this we are going to show multiplicity results for the problem under consideration for superlinear and sublinear growth, respectively.

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arXiv2024

Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results

In this work, we study the existence, non-existence, and uniqueness results for nonlocal elliptic equations involving logarithmic Laplacian, and subcritical, critical, and supercritical logarithmic nonlinearities. The Poho\u zaev's identity and D\'iaz-Saa type inequality are proved, which are of independent interest and can be applied to a larger class of problems. Depending upon the growth of nonlinearities and regularity of the weight function, we study the small-order asymptotic of nonlocal weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that the least energy solutions of a weighted nonlocal problem with superlinear or sublinear growth converge to a nontrivial nonnegative least-energy solution of Br\'ezis-Nirenberg type and logistic-type limiting problem respectively involving the logarithmic Laplacian.

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arXiv2024

Global gradient estimates for solutions of parabolic equations with nonstandard growth

We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equations of $p(x,t)$-Laplace type %with nonlinear sources depending on the solution and its gradient: \[ u_t-\Delta_{p(\cdot)}u=f(z)+F(z,u,\nabla u),\quad z=(x,t)\in Q_T=\Omega\times (0,T), \] with the nonlinear source $F(z,u,\nabla u)=a(z)|u|^{q(z)-2}u+|\nabla u|^{s(z)-2}(\vec c,\nabla u)$. It is proven the existence of a solution such that if $|\nabla u(x,0)|\in L^r(\Omega)$ for some $r\geq \max\{2,\max p(z)\}$, then the gradient preserves the initial order of integrability in time, gains global higher integrability, and the solution acquires the second-order regularity in the following sense: \[ \text{$|\nabla u(x,t)|\in L^r(\Omega)$ for a.e. $t \in (0,T)$}, \qquad \text{$|\nabla u|^{p(z)+\rho+r-2} \in L^1(Q_T)$ for any $\rho \in \left(0, \frac{4}{N+2}\right)$}, \] and \[ |\nabla u|^{\frac{p(z)+r}{2}-2}\nabla u\in L^2(0,T;W^{1,2}(\Omega))^N. \] The exponent $r$ is arbitrary and independent of $p(z)$ if $f\in L^{N+2}(Q_T)$, while for $f\in L^\sigma(Q_T)$ with $\sigma \in (2,N+2)$ the exponent $r$ belongs to a bounded interval whose endpoints are defined by $\max p(z)$, $\min p(z)$, $N$, and $\sigma$. An integration by parts formula is also proven, which is of independent interest.

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arXiv2023

Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=Ω\times (0,T)$, where $Ω\subset \mathbb{R}^N$, $N\geq 2$, is a $C^{2}$-smooth or convex bounded domain. It is assumed that $p\in C^{0,1}(\overline{Q}_T)$ is a given function, and that the nonlinear source $F(x,t,s, ξ)$ has a proper power growth with respect to $s$ and $ξ$. It is shown that if $p(x,t)>\frac{2(N+1)}{N+2}$, $f\in L^2(Q_T)$, $|\nabla u_0|^{p(x,0)}\in L^1(Ω)$, then the problem has a solution $u\in C^0([0,T];L^2(Ω))$ with $|\nabla u|^{p(x,t)} \in L^{\infty}(0,T;L^1(Ω))$, $u_t\in L^2(Q_T)$, obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: \[ \begin{split} & |\nabla u|^{2(p(x,t)-1)+r}\in L^1(Q_T)\quad \text{for any $0 < r < \frac{4}{N+2}$}, \\ & |\nabla u|^{p(x,t)-2} \nabla u \in W^{1,2}(Q_T)^N. \end{split} \]

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arXiv2023

On logarithmic double phase problems

In this paper we introduce a new logarithmic double phase type operator of the form\begin{align*}\mathcal{G}u:=-\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u+\mu(x)\left[\log(e+|\nabla u|)+\frac{|\nabla u|}{q(x)(e+|\nabla u|)}\right]|\nabla u|^{q(x)-2} \nabla u \right),\end{align*}where $\Omega\subseteq\mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $p,q\in C(\overline{\Omega})$ with $1<p(x)\leq q(x)$ for all $x\in\overline{\Omega}$ and $0\leq\mu(\cdot)\in L^1(\Omega)$. First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces $W^{1,\mathcal{H}_{\log}}(\Omega)$ and $W^{1, \mathcal{H}_{\log}}_0(\Omega)$ with $\mathcal{H}_{\log}(x,t)=t^{p(x)}+\mu(x)t^{q(x)}\log(e+t)$ for $(x,t)\in \overline{\Omega}\times [0,\infty)$ are separable, reflexive Banach spaces and $W^{1,\mathcal{H}_{\log}}_0(\Omega)$ can be equipped with an equivalent norm. We also prove several embedding results for these spaces and the closedness of these spaces under truncations. In addition we show the density of smooth functions in $W^{1,\mathcal{H}_{\log}}(\Omega)$ even in the case of an unbounded domain by supposing Nekvinda's decay condition on $p(\cdot)$. The second part is devoted to the properties of the operator and it turns out that it is bounded, continuous, strictly monotone, of type (S$_+$), coercive and a homeomorphism. As a result of independent interest we also present a new version of Young's inequality for the product of a power-law and a logarithm. In the last part of this work we consider equations driven by our new operator with superlinear right-hand sides. We prove multiplicity results for this type of equation, in particular about sign-changing solutions, by making use of a suitable variation of the corresponding Nehari manifold together with the quantitative deformation lemma and the Poincar\'e-Miranda existence theorem.

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arXiv2023

Semiclassical Moser-Trudinger inequalities

We extend the Moser-Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schr\"odinger operators on bounded domains.

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arXiv2022

A large class of nonlocal elliptic equations with singular nonlinearities

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$, with appropriate homogeneous boundary or exterior Dirichlet conditions. The operator $\mathbb{L}_γ^s$ belongs to a general class of nonlocal operators including typical fractional Laplacians such as restricted fractional Laplacian, censored fractional Laplacian and spectral fractional Laplacian. The nonlinear term $\mathcal{F}(u)$ covers three different amalgamation of nonlinearities: a purely singular nonlinearity $\mathcal{F}(u) = u^{-q}$ ($q>0$), a singular nonlinearity with a source term $\mathcal{F}(u) = u^{-q} + f(u)$, and a singular nonlinearity with an absorption term $\mathcal{F}(u) = u^{-q}-g(u)$. Based on a delicate analysis of the Green kernel associated to $\mathbb{L}_γ^s$, we develop a new unifying approach that empowered us to construct a theory for equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$. In particular, we show the existence of two critical exponents $q^{\ast}_{s, γ}$ and $q^{\ast \ast}_{s, γ}$ which provides a fairly complete classification of the weak-dual solutions via their boundary behavior. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.

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arXiv2022

Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^μ}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<μ<N$, $1<p<N$, $p<q<p+ \frac{αp}{N}$, $0 \leq a(\cdot)\in C^{0,α}(\mathbb{R}^N)$ with $α\in (0,1]$ and $f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.

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arXiv2022

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $Ω\times (0,T)$, where $Ω\subset \mathbb{R}^N$, $N\geq 2$, is a parallelepiped. The exponents of nonlinearity $p_i$ are given Lipschitz-continuous functions. It is shown that if $p_i(x,t)>\frac{2N}{N+2}$, \[ μ=\sup_{Q_T}\dfrac{\max_i p_i(x,t)}{\min_i p_i(x,t)}<1+\dfrac{1}{N}, \quad |D_{x_i}u_0|^{\max\{p_i(\cdot,0),2\}}\in L^1(Ω),\quad f\in L^2(0,T;W^{1,2}_0(Ω)), \] then the problem has a unique solution $u\in C([0,T];L^2(Ω))$ with $|D_{x_i} u|^{p_i}\in L^{\infty}(0,T;L^1(Ω))$, $u_t\in L^2(Q_T)$. Moreover, \[ |D_{x_i}u|^{p_i+r}\in L^1(Q_T)\quad \text{with some $r=r(μ,N)>0$},\qquad |D_{x_i}u|^{\frac{p_i-2}{2}}D_{x_i}u\in W^{1,2}(Q_T). \] The assertions remain true for a smooth domain $Ω$ if $p_i=2$ on the lateral boundary of $Q_T$.

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