Consider \begin{equation*}\ \left\{ \begin{aligned} -\mathcal_{\lambda,\Lambda}^{+}(D^{2}u)&=\mu u+a(x)u^{p}\quad\text{in}~\Omega,\\ u&=0\quad\text{on}~~\partial\Omega, \end{aligned} \right. \end{equation*} where $ \ Omega$ is a smooth bounded domain, $a$ is a continuous function which is allowed to change sign. Moreover, the set of zeros of $a$ may have a non-empty interior. The main objective of this article is to establish a uniform bound on the set of positive solutions of the above problem. This approach requires a suitable Liouville-type theorem. We first prove the Liouville-type theorem for $-\mathcal_{\lambda,\Lambda}^{+}(D^{2}v)=h(x_{1})f(v),$ where \(h\) is a strictly increasing one-dimensional weight and \(f\) is positive and nondecreasing.
Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation \begin{equation*} u''+a(t)u^\gamma=0 \end{equation*} with a two-step indefinite weight $a(t)$ have at most one positive solution for every $\gamma\in\mathbb{R}\setminus\left\{-1,0,1\right\}$. We prove this con...
We prove a Pogorelov interior estimate for strictly plurisubharmonic solutions of the complex Monge-Amp\`ere equation $\det(u_{i\bar{j}})=1$ with homogeneous Dirichlet data under the condition that, for some constant $\kappa\geq 1$, $(\kappa u_{i\bar{j}}-u_{is}u^{s\bar{t}}u_{\overline{tj}})$ is non-negative definite. F...
We study the Neumann boundary value problem $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda a(x)g(u) \quad \text{in } \Omega, \qquad \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\cdot \mathbf n = 0 \quad \text{on } \partial\Omega, $$ where $\Omega \subset \mathbb R^N$ is a bounded convex do...
In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition \begin{equation*} \begin{cases} -\Delta u = Ke^{2u}&\text{in } \mathbb{B}^{2},\\[2mm] \dfrac{\partial u}{\partial \nu}+\lambda = ke^{u}&\text{on } \partial\mathbb{B}^{2}, \end{cases} \end{equation*} where $K,k\in\mathbb{R}$, $\la...
In this paper, we study the nonlinear boundary value problem \begin{equation*} \begin{cases} -\Delta u=e^{2u},&\mbox{in } {\mathbb{D}},\\ \frac{\partial u}{\partial\nu}+\lambda=e^u ,&\mbox{on } {\mathbb{S}^{1}}, \end{cases} \end{equation*} where $\mathbb D$ is the unit disk, $\lambda$ is a constant and $\nu$ denotes th...
Given $m \in \mathbb{N} \setminus \{0\}$ and $\rho > 0$, we find solutions
$(\lambda,u)$ to the problem
\begin{equation*}
\begin{cases}
\biggl(-\dfrac{d^2}{d x^2}\biggr)^m u + \lambda G'(u) = F'(u),\\
\noalign{\vskip5pt}
\displaystyle
\int_{\mathbb{R}} K(u) dx = \rho,
\end{cases}
\end{equation*}
in the following cases...
Jacopo Schino, P. Smyrnelis· Topological Methods in Nonli...· 1 citation
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