Goal of this paper is to study the following singularly perturbed nonlinear Schrödinger equation $$ \varepsilon^{2s}(- \Delta)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N, $$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. When $\varepsilon>0$ is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of $V$; in particular, we improve the result in [37]. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting $s=1$ and $N\geq 3$, with an exponential decay of the solutions.

We consider the modified elliptic problem $$ -\Delta u-u\Delta u^2= a(x) u^\alpha+ \lambda b(x)u^{\beta}~ \mbox{in}~ \Omega, $$ with $u(x)=0$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^N$ is a regular domain and $N\geq3$, $0 < a(x)\in C(\Omega)\cap L^\infty(\Omega),b(x)\in C(\Omega)\cap L^\infty(\Omega)$, $0 < \alpha < 1 < \beta < \infty$ and $\lambda > 0$ is a parameter. By using sub- and super-solutions methods and variational methods, we establish the existence of two nontrivial solutions for the modified equation with appropriate exponents $\alpha,\beta$ and potentials $a(x), b(x)$.

The existence of solutions $u$ in $H^1(\mathbb R^N,\mathbb R^M)\cap H^2_{loc}(\mathbb R^N,\mathbb R^M)$ of a coupled semilinear system of second order elliptic partial differential equations on $\mathbb R^N$ of the form \[ \mathcal{P}[u] = f(x,u,\partial u), \quad x\in \mathbb R^N, \] under pointwise constraints is considered. The problem is studied via the constructed suitable topological invariant, the so-called constrained topological degree, which allows to get the existence of solutions of abstract problems considered as $L^2$-realizations of the approximating sequence of systems obtained by the truncation of the initial system to bounded subdomains. The key step of the proof consists in showing the relative $H^1$-compactness of the sequence of solutions to the truncated systems by the use of the so-called tail estimates. The constructions rely on the semigroup approach combined with topological methods, as well as invariance/viability techniques.

In this paper, we prove existence of a solution $(u,\varphi)$ in $W_0^{1,2}(\Omega) \times W_0^{1,2}(\Omega)$ for the Schrödinger-Maxwell type system $$ (*)\quad \left\{ \begin{array}{cl} -{\rm div} (M(x) \nabla u) + a(x)\,\varphi \,|u|^{r-2} u = f(x)\,, & \mbox{in $\Omega$,} \\ -{\rm div}(N(x) \nabla \varphi) = a(x) |u|^{r}\,, & \mbox{in $\Omega$,} \\ u = \varphi = 0\,, & \mbox{on $\partial\Omega$,} \end{array} \right. $$ with $M$ and $N$ bounded, uniformly elliptic matrices, $r > 1$, $a(x)$ in $L^{1}(\Omega)$ and $f$ such that $|f(x)| \leq Q\,a(x)$ for some positive constant $Q$.