We consider boundary value problems for elliptic differential-operator equations with the spectral parameter and operator-boundary conditions containing a parameter of the same order as the equation. An isomorphism and the corresponding estimate of a solution (with respect to the space variable and the parameter) are obtained. Then, discreteness of the spectrum and completeness of a system of root functions of the corresponding homogeneous problem are established. Finally, an application is given of the abstract results obtained on the isomorphism and the completeness to boundary value problems for elliptic differential equations with a parameter in non-smooth domains.

We consider the system $$ J\dot u=\nabla { \mathcal {H}}(u)+f(u)+p(t)\,, $$ where ${ \mathcal {H}}:{{\mathbb R}} ^2\to{{\mathbb R}} $ is of class $C^1$ with locally Lipschitz continuous gradient, $f:{{\mathbb R}} ^2\to{{\mathbb R}} ^2$ is locally Lipschitz continuous and bounded, and $p:{{\mathbb R}} \to{{\mathbb R}} ^2$ is measurable, bounded and $T-$periodic. Here, $J= \begin{pmatrix} \scriptstyle 0 & \!\!\!\!\scriptstyle -1 \cr \scriptstyle 1 & \scriptstyle \!0 \end{pmatrix} $ is the standard symplectic matrix. For some classes of functions $f,$ we give new existence theorems for periodic solutions and for unbounded solutions. Applications are given to forced second-order differential equations with separated nonlinearities.

We prove existence results for complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory, developed in [1, 2, 5].

We study existence and nonexistence of least energy solutions of a quasilinear critical growth equation with degenerate $m$-Laplace operator in a bounded domain in $\mathbb{R}^n$ with $n>m>1$. Existence and nonexistence of solutions of this problem depend on a lower order perturbation and on the space dimension $n$. Our proofs are obtained with critical point theory and the lack of compactness, due to critical growth condition, is overcome by constructing minimax levels in a suitable compactness range.