Related papers: On Choquard-Kirchhoff Type Critical Multiphase Pro…

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution,…

IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u=(I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array}…

In the present paper, we consider the nonlocal Kirchhoff problem \begin{eqnarray*} -\left(\epsilon^2a+\epsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where…

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type $$ -M\left(\displaystyle\int_\Omega \Phi(|\nabla u|)dx\right)\Delta_\Phi u = f(u) \ \ \mbox{in} \ \ \Omega, \ \ u=0 \ \…

This paper is concerned with the quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality. Namely, we give quantitative estimates for the Choquard equation: $$-\Delta u=(I_{\mu}\ast|u|^{2_\mu^*}) u^{2_\mu^*-1}\ \…

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general…

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: $-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }\Omega,$ and $u=0,\text{ on }\partial…

We investigate the degenerate fractional Schr\"{o}dinger-Kirchhoff-Poisson equation in $\mathbb{R}^3$ with critical nonlinearity and electromagnetic fields $\varepsilon^{2s} M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u + \phi u = k(x)|u|^{r-2}u +…

In this note we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-\Delta_p u = |u|^{p^*-2}u + \lambda f(x,u)$ in a smooth bounded domain $\Omega$ of $\R^N$ with homogeneous Dirichlet…

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…

We consider the following critical weakly coupled elliptic system \[ \begin{cases} -\Delta u_i = \mu_i |u_i|^{2^*-2}u_i + \sum_{j \neq i} \beta_{ij} |u_j|^{\frac{2^*}{2}} |u_i|^{\frac{2^*-4}{2}} u_i & \text{in $\Omega_\varepsilon$} u_i >0 &…

In this paper, we study the following quasilinear Schr\"{o}dinger equation of Choquard type $$ -\triangle u+V(x)u-\triangle (u^{2})u=(I_\alpha *|u|^p)|u|^{p-2}u, \ \ x \in \mathbb{R}^{N}, $$ where $N\geq 3$,\ $0<\alpha<N$,…

This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*}…

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 establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-\Delta u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\ $-\Delta…

We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold…

We consider the following nonlocal Br\'ezis-Nirenberg type critical Choquard problem involving the Grushin operator \begin{equation*} \left\{ \begin{aligned} -\Delta_\gamma & u =\lambda u + \left(\displaystyle\int_\Omega…

We study the Cauchy problem for the nonlinear Schr\"{o}dinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We…

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll}…

Using variational methods, we obtain several multiplicity results for double phase problems that involve variable exponents and a new type of critical growth. This new critical growth is better suited for double phase problems when compared…