相关论文: Exact multiplicity of solutions for some semilinea…

In this paper we investigate the existence of multiple solutions for the following two fractional problems \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\Omega})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \partial…

We prove a multiplicity result for \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta_g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\[1mm] -\Delta_g \phi +a^{2}\Delta_g^{2} \phi + m^2 \phi =4\pi u^{2} \end{cases} \text{ in }M, \end{equation*}…

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -\Delta u= h(u){f} \ \ \text{in}\,\ \Omega, $$ where $f$ is an irregular datum,…

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function…

This paper is concerned with existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, precise lower bound estimates of…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$:…

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and…

In this paper we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi type problem $$ \left\{ \begin{array}{lcl} -\Delta u=P(x)\Big( g(u)+f(x)\Big) \mbox{ in } \mathbb{R}^N,\\ u \in D^{1,2}(\R^N),\…

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal…

In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) & \text{in } \Omega, \frac{\partial u}{\partial \nu} + \gamma u=0 & \text{on }…

We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -\Delta_1 u +g(u)|Du|=h(u)f & \text{in $\Omega$,} \\ u=0 & \text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$,…

In this paper, we study the existence of nontrivial solutions of the Dirichlet boundary value problem for the following elliptic system: \begin{equation} \left\{ \begin{aligned} -\Delta u & = au + bv + f(x,u,v); &\quad\mbox{ for…

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\R^n\setminus\Omega$, for some…

In order to obtain solutions to problem $$ {{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen…

Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $\Delta_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of…

We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where…

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show…