This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear prima facie alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element...

Let $\Omega \subset {ℝ}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem $$\begin{array}{c}\hfill -{\Delta }_{p}u-{\Delta }_{q}u=\lambda \left(𝐱\right){\left|u\right|}^{p^{☆}-2}u+\mu {\left|u\right|}^{r-2}u\end{array}$$ where $\mu$ is a positive parameter, $1<q\le p<n$, $r\ge p^{☆}$ and $p^{☆}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p,q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

We provide a detailed analysis of the minimizers of the functional $u\mapsto {\int }_{{ℝ}^n}{\left|\nabla u\right|}^2+D{\int }_{{ℝ}^n}{\left|u\right|}^{\gamma }$, $\gamma \in (0,2)$, subject to the constraint ${\parallel u\parallel }_{L^2}=1$. This problem,e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties...

We provide a detailed analysis of the minimizers of the functional $u\mapsto {\int }_{{ℝ}^n}{\left|\nabla u\right|}^2+D{\int }_{{ℝ}^n}{\left|u\right|}^{\gamma }$, $\gamma \in (0,2)$, subject to the constraint ${\parallel u\parallel }_{L^2}=1$. This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties...

We present some results concerning the problem $-\Delta u=\lambda \frac{u}{{\left|x\right|}^2}+u^q$, $u>0$ in $\Omega$, ${u|}_{\partial \Omega }=0$, where $0<q<(N+2)/\left(N-2\right)$, $q\ne 1$, $\lambda \ge 0$ and $\Omega$ is a smooth bounded domain containing the origin. In particular, bifurcation and uniqueness results are discussed.