Related papers: Mixed order elliptic problems driven by a singular…

In this paper, we study the following Hamiltonian Choquard-type elliptic systems involving singular weights \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\Delta u + V(x)u = \Big(I_{\mu_{1}}\ast…

In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left\{\begin{array}{cc} &-M(\varphi_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla…

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a…

In the present paper, we study a class of quasilinear Choquard equations involving $N$-Laplacian and the nonlinearity with the critical exponential growth. We discuss the existence of positive solutions of such equations.

We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where…

In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$…

We consider the planar logarithmic Choquard equation $$- \Delta u + a(x)u + (\log|\cdot| \ast u^2)u = 0,\qquad \text{in } \mathbb{R}^2$$ in the strongly indefinite and possibly degenerate setting where no sign condition is imposed on the…

We prove existence of a positive radial solution to the Choquard equation $$-\Delta u +V u=(I_\alpha\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,\Omega$$ with Neumann or Dirichlet boundary conditions, when $\Omega$ is an annulus, or an…

This paper deals with the existence of multiple solutions for the quasilinear equation $-\mathrm{div}\,\mathbf{A}(x,\nabla u)| u| ^{\alpha (x)-2}u=f(x,u)$ in $ \mathbb{R} ^{N}$, which involves a general variable exponent elliptic operator…

We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $\mathcal Au+\Phi(x,u,\nabla u)=\mathfrak{B}u+f$ in $\Omega$, where $\Omega$ is a bounded open subset of $\mathbb R^N$ and…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we…

We consider a Dirichlet elliptic problem driven by the Laplacian with singular and superlinear nonlinearities. The singular term appears on the left-hand side while the superlinear perturbation is parametric with parameter $\lambda>0$ and…

In the present work we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $\mathbb{R}^2$, for $ a>0 $, $ \lambda >0 $ and a nonlinearity $f$ with exponential critical growth.…

In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} \Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the…

In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation \begin{equation*} (-\Delta)^{\frac{\alpha}{2}} u=\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{\alpha}-2}u, \quad\text{in}\,\,\Omega,…

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{…

We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-\Delta u+V(x)u=\left(I_\alpha\ast |u|^p\right)|u|^{p-2}u+\lambda u\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right.…

We study the Choquard equation involving mixed local and nonlocal operators $$-\Delta u+(-\Delta)^{s}u+V(x)u=(\frac{1}{|x|^{\mu}}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $\mu\in(0,2)$, $F(t)=\int_{0}^{t} f(\tau)\,d\tau$, and…

In this paper, we study the mass-constrained fractional Choquard equation \( (-\Delta)^s u = \lambda u + \alpha (I_\mu * |u|^{\frac{2N-\mu}{N}})|u|^{\frac{2N-\mu}{N}-2}u + (I_\mu * |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the…