Search

This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another notable feature is that the source terms in both the purely singular and perturbed components can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent., Comment: 38 pages, comments are welcome

In this article, we establish variational characterization of solutions of a mixed local and nonlocal singular semilinear elliptic problem. As an application of this fact, we deduce decomposition property. Further, using moving plane method, we prove symmetry of solutions of a mixed local and nonlocal singular perturbed problem, where the decomposition plays a crucial role. Finally, we apply the variational property to deduce existence, nonexistence and regularity results of a mixed local and nonlocal singular jumping problem., Comment: 32 pages. Comments are welcome

This article is twofold. In the first part of the article, we consider Brezis-Oswald problem involving the mixed anisotropic and nonlocal $p$-Laplace operator and establish existence, uniqueness, boundedness and strong maximum principle. Further, for some mixed anisotropic and nonlocal $p$-Laplace type equations, we obtain Sturmian comparison theorem, prove some comparsion results, nonexistence results, establish weighted Hardy type inequality and study a system of singular mixed anisotropic and nonlocal $p$-Laplace equations. One of the key ingredient is the combination of Picone identity already available in the separate local and nonlocal cases. In the second part of the article, we are mainly concerned with regularity estimates. To this end, in the elliptic setting, we establish weak Harnack inequality and semicontinuity results. Further, we consider a class of doubly nonlinear mixed anisotropic and nonlocal equations and prove semicontinuity results and pointwise behavior of solutions. These are based on suitable energy estimates and De Giorgi type lemmas and expansion of positivity. Finally, we establish various energy estimates which may be of independent interest., Comment: 51 pages, comments are welcome

In this paper, we are concerned with the H\"older regularity for solutions of the nonlocal evolutionary equation $$ \partial_t u+(-\Delta_p)^s u = 0. $$ Here, $(-\Delta_p)^s$ is the fractional $p$-Laplacian, $0

IIn this article, we consider the weighted Steklov eigenvalue problem and the weighted Schr\"odinger--Steklov eigenvalue problem in outward cuspidal domains. We prove solvability of this spectral problem both in the linear and non-linear case., Comment: 15 pages

In this paper, we will study Neumann $(p,q)$-eigenvalue problem for the weighted $p$-Laplace operator in outward H\"older cuspidal domains. The suggested method is based on the composition operators on weighted Sobolev spaces., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2301.11037

We study the fractional $p$-Laplace equation $$ (-\Delta_p)^s u = 0 $$ for $0

We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions., Comment: 26 pages, comments are welcome

In this article, we consider the spectral problem for the mixed local and nonlocal $p$-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet $(p,q)$-eigenvalue problem in a bounded domain $\Omega\subset\mathbb{R}^N$ under the assumption that $1

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded H\"older $\gamma$-singular domains $\Omega_{\gamma}\subset \mathbb{R}^n$. In the case $1

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish existence of at least one weak solution for a parameter dependent singular nonlinearity and existence of multiple solution for purturbed singular nonlinearity. Our argument is based on the variational and approximation approach., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:2112.13294

We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and H\"older continuity with an explicit H\"older exponent in the general case. For certain parameters, our results also imply H\"older continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method., Comment: 39 pages, references are updated

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(\xi,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $\xi$, and measurable with respect to $X,Y$ and $t$. $A=A(\xi,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{\"o}lder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded $X$, $Y$ and $t$ dependent domains., Comment: 34 Pages

This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case is even new in the classical $p$-Laplacian case., Comment: 28 pages, comments are welcome

We consider a class of singular weighted anisotropic $p$-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case., Comment: 20 pages, comments are welcome

In this article we study singular subelliptic $p$-Laplace equations and best constants in Sobolev inequalities on nilpotent Lie groups. We prove solvability of these subelliptic $p$-Laplace equations and existence of the minimizer of the corresponding variational problem. It leads to existence of the best constant in the corresponding $(q,p)$-Sobolev inequality, $0

In this article, we consider a class of degenerate singular problems. The degeneracy is captured by the presence of a class of $p$-admissible weights, which may vanish or blow up near the origin. Further, the singularity is allowed to vary inside the domain. We provide sufficient conditions on the weight function, on the singular exponent and the source function to establish regularity and existence results., Comment: 15 pages

In the article we study the Dirichlet $(p,q)$-eigenvalue problem for subelliptic non-commutative operators on nilpotent Lie groups. We prove solvability of this eigenvalue problem and existence of the minimizer of the corresponding variational problem., Comment: 10 pages

We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $\Omega\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_{\Omega}|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_{\Omega}\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(\Omega,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(\Omega,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $\Omega$. We discuss the case $0

We establish multiplicity results for the following class of quasilinear problems $$ \left\{ \begin{array}{l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \leqno{(P)} $$ where $\Delta_{\Phi}u=\text{div}(\varphi(x,|\nabla u|)\nabla u)$ for a generalized N-function $\Phi(x,t)=\int_{0}^{|t|}\varphi(x,s)s\,ds$. We consider $\Omega\subset\mathbb{R}^N$ to be a smooth bounded domain that contains two disjoint open regions $\Omega_N$ and $\Omega_p$ such that $\overline{\Omega_N}\cap\overline{\Omega_p}=\emptyset$. The main feature of the problem $(P)$ is that the operator $-\Delta_{\Phi}$ behaves like $-\Delta_N$ on $\Omega_N$ and $-\Delta_p$ on $\Omega_p$. We assume the nonlinearity $f:\Omega\times\mathbb{R}\to\mathbb{R}$ of two different types, but both behaves like $e^{\alpha|t|^\frac{N}{N-1}}$ on $\Omega_N$ and $|t|^{p^*-2}t$ on $\Omega_p$ as $|t|$ is large enough, for some $\alpha>0$ and $p^*=\frac{Np}{N-p}$ being the critical Sobolev exponent for $1

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse H\"older inequality and a logarithmic estimate for weak supersolutions., Comment: 27 pages

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -\Delta_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }\Omega,\\ u&>0\text{ in }\Omega,\,u=0\text{ on }\partial\Omega, \end{split}\right. \end{equation*} under the assumption $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\la>0$ and $0

We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local H\"older continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation., Comment: 33 pages, Comments are welcome

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case and in the nonlocal elliptic case to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse H\"older inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions., Comment: 32 pages

In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincar\'e inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting., Comment: 12 pages, Comments are welcome

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }\Omega. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\frac{1}{u}$ in a bounded smooth domain $\Omega$ and nonexistence results for $f(u)=-e^\frac{1}{u}$ or $-(u^{-\delta}+u^{-\gamma}),$ $\delta,\gamma>0$ with $\Omega=\mathbb{R}^N$ respectively., Comment: 18 pages

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u^\delta}\,\,\text{ in }\,\,\Omega, u>0\text{ in }\Omega,\\ u = 0 \text{ on } \partial\Omega, \end{cases} \end{gather*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 2$, $w$ belong to the Muckenhoupt class $A_p$ for some $10$., Comment: 24 pages

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$., Comment: Opuscula Mathematica, 2020

In this paper we present some non existence results concerning the stable solutions to the equation $$\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=g(x)f(u)\;\;\mbox{in}\;\;\mathbb{R}^N;\;\;p\geq 2$$ when $f(u)$ is either $u^{-\delta}+u^{-\gamma}$, $\delta,\gamma>0$ or $\exp(\frac{1}{u})$ and for a suitable class of weight functions $w,g$.

We provide sufficient conditions on $w\in L^1_{loc}(\mathbb{R}^N)$ such that the weighted $p$-Laplace equation $$-\operatorname{div}\big(w(x)|\nabla u|^{p-2}\nabla u\big)=f(u)\;\;\mbox{in}\;\;\mathbb{R}^N$$ does not admit any stable $C^{1,\zeta}_{loc}$ solution in $\mathbb{R}^N$ where $f(x)$ is either $-x^{-\delta}$ or $e^x$ for any $0<\zeta<1$.

For an open, bounded domain $\Om$ in $\mathbb{R}^N$ which is strictly convex with $C^2$ boundary, we show that there exists a $\land>0$ such that the singular quasilinear problem \begin{eqnarray*} &-\delp u =\cfrac{\lambda}{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atleast two solution $ u$ and $v$ in $W^{1,p}_{loc}(\Om)\cap L^{\infty}(\Om)$ for any $\del>0$ and $0<\lam<\land$ provided $10$.