We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $-\Delta u=f_{\lambda }(x,u)$, $u\in H_0^1\left(\Omega \right)$, where $\Omega$ is a bounded domain in ${ℝ}^N$, $N\ge 3$ and $\lambda >0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $\lambda a\left(x\right)u^q+b\left(x\right)u^p$, where $0\le q<1<p\le 2^{*}-1$. The coefficient $a\left(x\right)$ is assumed to be nonnegative but $b\left(x\right)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this...

We obtain sufficient conditions for nonexistence of nontrivial solutions for some classes of nonlinear partial differential inequalities containing the fractional powers of the Laplace operator.

In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

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.

The Dirichlet boundary value problem for systems of elliptic partial differential equations at resonance is studied. The existence of a unique generalized solution is proved using a new min-max principle and a global inversion theorem.

We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.