An elliptic problem involving critical Choquard and singular discontinuous nonlinearity

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying variational methods and using the notion of generalized gradients for Lipschitz continuous functional, we obtain the existence and the multiplicity of weak solutions for some suitable range of $\lambda$ and $\gamma$. Finally by studying the $L^\infty$-estimates and boundary behavior of weak solutions, we prove their H\"{o}lder and Sobolev regularity

Choquard equation involving mixed local and nonlocal operators

In this article, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects and in the presence of critical nonlinearity of Hartree type. To this end, we first investigate the corresponding Hardy-Littlewood-Sobolev inequality and detect the optimal constant. Using variational methods and a Poho\v{z}aev identity we then show the existence and nonexistence results for the corresponding subcritical perturbation problem

Variational Inequalities for the Fractional Laplacian

In this paper we study the obstacle problems for the fractional Lapalcian of order s 08 (0, 1) in a bounded domain, under mild assumptions on the data

A note on the global regularity results for strongly nonhomogeneous p,qp,q-fractional problems and applications

In this article, we communicate with the glimpse of the proofs of new global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-\Delta )^{s_1}_{p}+(-\Delta )^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $

A note on the global regularity results for strongly nonhomogeneous p,qp,q-fractional problems and applications

In this article, we communicate with the glimpse of the proofs of global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We also obtain the boundary H\"older continuity results for the weak solutions to the corresponding problems involving at most critical growth nonlinearities. These results are almost optimal. Moreover, we establish Hopf type maximum principle and strong comparison principle. As an application to these new results, we prove the Sobolev versus H\"older minimizer type result, which provides the multiplicity of solutions in the spirit of seminal work \cite{Brezis-Nirenberg}

Symmetry of solutions to singular fractional elliptic equations and applications

In this article, we study the symmetry of positive solutions to a class of singular semilinear elliptic equations whose prototype is \begin{align*} (P) \quad \left\lbrace \begin{array}{ll} (-\Delta )^{s}u = \frac{1}{u^\delta } + f(u), \; u>0\quad & \text{ in }\Omega ; \\ u=0 & \text{ in } \mathbb{R}^n\setminus \Omega ,\\ \end{array} \right. \end{align*} where $00$, $f(u)$ is a locally Lipschitz function. We prove that classical solutions are radial and radially decreasing (see Theorem 1). The proof uses the moving plane method adapted to the non local setting. We then give two applications of this main result: Theorem 2 establishes the uniform apriori bound for classical solutions in case of polynomial growth nonlinearities whereas Theorem 3 ensures in case of exponential growth nonlinearities the convergence of large solutions with unbounded energy to a singular solution

Symmetry of solutions to singular fractional elliptic equations and applications

In this article, we study the symmetry of positive solutions to a class of singular semilinear elliptic equations whose prototype is \begin{align*} (P) \quad \left\lbrace \begin{array}{ll} (-\Delta )^{s}u = \frac{1}{u^\delta } + f(u), \; u>0\quad & \text{ in }\Omega ; \\ u=0 & \text{ in } \mathbb{R}^n\setminus \Omega ,\\ \end{array} \right. \end{align*} where $00$, $f(u)$ is a locally Lipschitz function. We prove that classical solutions are radial and radially decreasing (see Theorem 1). The proof uses the moving plane method adapted to the non local setting. We then give two applications of this main result: Theorem 2 establishes the uniform apriori bound for classical solutions in case of polynomial growth nonlinearities whereas Theorem 3 ensures in case of exponential growth nonlinearities the convergence of large solutions with unbounded energy to a singular solution