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 semilinear heat equations. For $\frac{n}{n-2}<q<\frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.
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 $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.
Rubén Fiñana, A. Molino· Nonlinear Analysis· 0 citations
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 $u\in\tilde H^s(\Omega)$ changes sign in $\Omega$.
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 Hausdorff measure, we show that $u$ is a weak solution of these nonlinear wave equations in the entire $\mathcal{X}.$
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, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.
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 $\Omega$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $\Omega$ is a ball, the problem is open for arbitrary $\beta>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $\beta>0$ provided that $p$ and $\Omega$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.
Mengyao Chen, Massimo Grossi, Qi Li· 1 citation
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
