Nontrivial solutions of variational inequalities. The degenerate case

We consider a class of asymptotically linear variational inequalities. We show the existence of a nontrivial solution under assumptions which allow the problem to be degenerate at the origin

Existence of nontrivial solutions for semilinear problems with strictly differentiable nonlinearity

The existence of a nontrivial solution for semilinear elliptic problems with strictly differentiable nonlinearity is proved. A result of homological linking under nonstandard geometrical assumption is also shown. Techniques of Morse theory are employed

Lagrangian systems with Lipschitz obstacle on manifolds

Lagrangian systems constrained on the closure of an open subset with Lipschitz boundary in a manifold are considered. Under suitable assumptions, the existence of infinitely many periodic solutions is proved

Perturbations of critical values in nonsmooth critical point theory

* Supported by Ministero dell’Università e della Ricerca Scientifica e Tecnologica (40% – 1993). ** Supported by Ministero dell’Università e della Ricerca Scientifica e Tecnologica (40% – 1993).The perturbation of critical values for continuous functionals is studied. An application to eigenvalue problems for variational inequalities is provided

Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

The paper deals with the existence of positive solutions with prescribed $L2$ norm for the Schrodinger equation $$-\Delta u+\lambda u+V(x)u=|u|{p-2}u,\quad u\in H1_0(\Omega),\quad\int_\Omega u2{\rm d}\,x=\rho2,\quad\lambda\in\mathbb{R},$$ where $\Omega =\mathbb {R}N$ or $\mathbb {R}N\setminus \Omega$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (D?), or if $\|V\|_{Lq}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (D?) is fulfilled. There are no assumptions on the size of $\mathbb {R}N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}N$

Nontrivial solutions of p-superlinear p-Laplacian problems via a cohomological local splitting

We consider a quasilinear equation, involving the p-Laplace operator, with a p-superlinear nonlinearity. We prove the existence of a nontrivial solution, also when there is no mountain pass geometry, without imposing a global sign condition. Techniques of Morse theory are employed