Related papers: Supercritical problems on manifolds

In this paper we are concerned with the following nonlinear Choquard equation $$-\Delta u+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $N\geq4$,…

We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega,…

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2, \ \…

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \mathrm{div}_g u &= 0. \end{align*} We show that any quadratic ODE $\partial_t y =…

We provide a sufficient condition for the existence of a positive solution to $-\Delta u+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and…

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward…

We provide a multiplicity result for solutions of time-independent Gross-Pitaevskii equations on closed Riemannian manifolds. Such solutions arise as (possibly non-minimizing) critical points of the Ginzburg-Landau energy having prescribed…

We analyze the behavior of families $(u_\alpha)_{\alpha>0}$ of solutions to the high-order critical equation $P_\alpha u_\alpha=\Delta_g^k u_\alpha +\hbox{lot}=|u_\alpha|^{2^\star-2}u_\alpha$ on a Riemannian manifold $M$, with a uniform…

In this paper, we show that for any finite subgroup $\Gamma < O(4)$ acting freely on $\mathbb{S}^3$, there exists a $4$-dimensional complete Riemannian manifold $(M,g)$ with ${\rm Ric}_g \geq 0 $, such that the asymptotic cone of $(M,g)$ is…

Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of…

In this paper, we continue to construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure now is carried out by constructing solutions to the…

We consider the problem $$ (P_\lambda)\quad -\Delta_{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }\Omega $$ under Dirichlet or Neumann boundary conditions. Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$…

We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $ -\Delta u + F(u) = f $ with Dirichlet or Neumann boundary conditions on a precompact domain $ \Omega \subset…

The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e.…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and $x_0 \in M$ be an isolated local minimum of the scalar curvature $s_g$ of $g$. For any positive integer $k$ we prove that for $\epsilon >0$ small enough the subcritical…

This paper deals with a fourth order elliptic equation on compact Riemannian manifolds.We establish the existence of solutions to the equation with critical Sobolev growth which is the subject of the first theorem. In the second one, we…

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\,…

In this paper we consider the nonlinear Choquard equation $$ -\Delta u+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<\mu<N$, $N\geq3$, $g(x,u)$ is…

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $\Omega$ and $u= 0$ on…