Related papers: Existence of Kazdan-Warner equation with sign-chan…

We consider in 2D the following special case of the Mumford-Shah functional $$ J(u, \Gamma)=\int_{B_1\backslash\Gamma} |\nabla u|^2 dx + \lambda^2 \frac{\pi}{2} \mathcal{H}^1(\Gamma). $$ It is known that if the minimizer has a crack-tip in…

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u = f.u^{\frac{n+2}{n-2}}$ on a compact riemannian manifold in the limit case normally not solved by variational…

For a contravariant 4-metric which changes signature from Lorentzian to Riemannian across a spatial hypersurface, the mixed Einstein tensor is manifestly non-singular. In Gaussian normal coordinates, the metric contains a step function and…

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,…

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-\Delta)^{\frac{\alpha}{2}} u+\lambda\, \Delta u \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $\alpha\in (0,2)$, $\lambda\in\mathbb{R}$…

Let $(M,g)$ be a closed Riemannian manifold of dimension at least $3$. Let $S$ be the union of the focal submanifolds of an isoparametric function on $(M,g)$. In this article we address the existence of solutions of the Hardy-Sobolev type…

In this paper we consider the following class of elliptic problems $$- \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,\,\, x\in \mathbb{R}^N$$ where $1<q<2<p<2^*-1= \frac{N+2}{N-2}$, $a_{\lambda}(x)$ is a sign-changing…

Given $\mu > 0$, we study the elliptic problem: \begin{align*} \text{ find } (u,\lambda) \in H_0^1(\Omega) \times \mathbb{R} \text{ such that } -\Delta u + \lambda u = |u|^{p-2}u \text{ in } \Omega \text{ and } \int_\Omega|u|^2dx = \mu,…

Let $G=(V,E)$ be a finite connected graph, and let $\kappa: V\rightarrow \mathbb{R}$ be a function such that $\int_V\kappa d\mu<0$. We consider the following Kazdan-Warner equation on $G$:\[\Delta u+\kappa-K_\lambda e^{2u}=0,\] where…

It is well known that minimizers of the Allen-Cahn-type functional \[ J_\epsilon(u):=\int_\Omega\frac{\epsilon|\nabla u|^2}{2}+\frac{W(u)}{\epsilon}, \] where $W$ is a double-well potential, resemble minimal surfaces in the sense that their…

This paper is concerned with the existence of sign-changing solutions to non local Kirchhoff type problems of the form \begin{equation}\label{s}\tag{S} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\, \text{ in }\Omega,\quad\quad…

We consider the problem $$(P_\eps)\qquad \Delta u +\lambda {e^{u}\over\int\limits_{\Omega\setminus B(\xi,\eps)}e^u}=0\ \hbox{in}\ \Omega\setminus B(\xi,\eps),\quad u =0\ \hbox{on}\ \partial\(\Omega\setminus B(\xi,\eps)\), $$ where $\Omega$…

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and…

We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent: \begin{align*}\left\{\begin{array}{l l} { (-\Delta)^{s}u+ \lambda u= \abs{u}^{p-1}u } & \text{in $ \Omega,$…

Consider the nonlinear heat equation v_t-\Delta v=|v|^{p-1}v in the unit ball of R^2, with Dirichlet boundary condition. Let u_{p,K} be a radially symmetric, sign-changing stationary solution having a fixed number K of nodal regions. We…

In this paper we consider the prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds. By deducing the expression of the Gauduchon scalar curvature under the conformal variation, the problem is reduced to solve a…

We consider the Brezis-Nirenberg problem: $$-\Delta u =\lambda u + |u|^{p-1}u\qquad \mbox{in}\,\, \Omega,\quad u=0\,\, \mbox{on}\,\,\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$,…

As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a $T-$periodic solution to the second order differential equation \begin{equation*} u"=\frac{h(t)}{u^{\lambda}} \end{equation*} are…

We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{ in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded regular…

This paper is devoted to the existence of multiple sign-changing solutions of prescribed mass for a mass-supercritical nonlinear Schr\"odinger equation set on a compact metric graph. In particular, we obtain, in the supercritical mass…