In this paper, on a bounded domain $\Omega\subset {\bf R}^n$, we consider a nonlocal problem of the type $$\cases{-\Delta u=Q(u,\lambda)f(u)&in $\Omega$ \cr&\cr u=0&on $\partial\Omega$\cr}$$ where $Q:H^1_0(\Omega)\times {\bf R}\to {\bf R}$, proving, under suitable assumptions, the existence of at least three weak solutions for each $\lambda$ running in a suitable interval.
In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of...
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in(1,\frac 32)$ the quadratic form $Q^{\rm SP}_s[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto|u|$ provided $u\in\tilde H^s(\Om...
Egor Ignatev, A. Nazarov, Pavel Nichitenko et al.· 0 citations
We prove Radó theorem for nonlinear wave equations. Considering $u$ as a locally Lipschitz continuous function on an open set $\mathcal{X}$ in $\mathbb{R}^{n+1}$ to be a weak solution of the equations $ \partial_{tt}u-div(|u’|^{p-2}u’)=0$, on $\mathcal{X}\backslash u^{-1}(0)$, where $p \ge 2$, and employing the Hausdor...
M. Ouedraogo, T. Ouedraogo, B. Kabore et al.· Far East Journal of Mathemat...· 0 citations
In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory,...
In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -\Delta u = u^p,&\text{in } \Omega,\\ u>0,&\text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0,&\text{on } \partial \Omega, \end{cases} $$ where $\beta>0$, $p$ is subcritical, and $\Omeg...
Mengyao Chen, Massimo Grossi, Qi Li· 1 citation
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