In this paper, we study the nonhomogeneous elliptic equations $-\Delta u(x)+u(x)=\lambda (f(x,u)+h(x))$ in ${\mathbb R}^{N}$, where $f(x,u)$ and $h(x)$ satisfy some assumptions. We use variational methods to obtain the existence results and give a clever argument to establish the exact number of solutions. In addition to a lack of compactness the main difficulty to overcome is the degenerated structure of the set of possible critical points.

We study the exact structure of positive solutions of an Ambrosetti-Brezis-Cerami problem and its generalization in one space variable and in the classical Laplacian case. We prove the exact multiplicity result when $\lambda $ ranges over the whole interval $\left( 0,\infty \right) $ and get more detailed results of the solution curve. The proof of our exact multiplicity result uses the modified time-map techniques which can be adapted, and the exact multiplicity result can be extended to a more general $k$-Laplacian problem with $k>1.$

In this note we prove some existence and nonexistence results for a semilinear equation of the fourth order involving an exponential function via variational techniques. Such equations may occur while studying traveling waves in suspension bridges.

In this paper we give, for the first time, an abstract interpretation of initial--boundary-value problems for hyperbolic equations such that a part of the boundary-value conditions contains also a differentiation of the time $t$ of the same order as the equations. Initial--boundary-value problems for hyperbolic equations are reduced to the Cauchy problem for a system of hyperbolic differential-operator equations. A solution of this system is not a vector function but one function. At the same time, the system is not overdetermined. We prove the well-posedness of the Cauchy problem, and for some special cases we give an expansion of a solution to the series of eigenvectors. As application we show, in particular, a generalization of the classical Fourier method of separation of variables.

We prove optimal embedding estimates for the domains of second-order elliptic operators in $L^1$ spaces. Our procedure relies on general semigroup theory and interpolation arguments, and on estimates for $\nabla T(t)f$ in $L^1$, in $L^\infty$, and possibly in fractional Sobolev spaces, for $f\in L^1$. It is applied to a number of examples, including some degenerate hypoelliptic operators, and operators with unbounded coefficients.

Linear parabolic equations with coefficients of the lower-order terms unbounded, and with a small parameter multiplying some of the second (highest) space derivatives are considered, in the limiting case when such a parameter goes to zero. This yields a degenerate parabolic (ultraparabolic) equation with one space-like variable, $x$, and two time-like variables, $y$ and $t$. No boundary-layer is found to be needed in the case of the boundary-value problem on the $x$-unbounded domain $\mathcal{Q}_T=\{(x,y,t)\in \mathbb{R}\times [0,1]\times[0,T]\}$ with a periodic boundary condition in the variable $y$ and initial data at $t=0$.

A class of semilinear elliptic equations with dependence on the gradient is considered. The existence of a positive and a negative solution is stated through an iterative method based on mountain-pass techniques.

We study scattering theory for nonlinear Schrödinger equations with cubic and quadratic nonlinearities in one and two space dimensions, respectively. For example, the nonlinearities are the sum of the gauge-invariant term and non-gauge-invariant terms such as $\lambda_0 \!|u|^2u +\lambda_1 u^3 +\lambda_2 u\bar{u}^2 +\lambda_3 \bar{u}^3$ in the one-dimensional case, where $\lambda_0 \in {\mathbb R}$ and $\lambda_1,\lambda_2,\lambda_3$ $ \in {\mathbb C}$. The scattering theory for these equations belongs to the long-range case. We show the existence and uniqueness of global solutions for those equations which approach a given modified free profile. The same problem for the nonlinear Schrödinger equation with the Stark potentials is also considered.

In this paper an initial--boundary-value problem for a nonlinear string (or wave) equation with nonclassical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the (positive) damping is generated. Existence, uniqueness, and regularity of solutions to this problem are investigated.

We generalize the sub-supersolution method known for weak solutions of single and multivalued equations to quasilinear elliptic hemivariational inequalities. To this end we first introduce our basic notion of sub- and supersolutions on the basis of which we then prove existence, comparison, compactness, and extremality results for the hemivariational inequalities under considerations.

The Klein-Gordon-Schrödinger system with Yukawa coupling is shown to have a unique global solution for rough data, which do not necessarily have finite energy. The proof uses a generalized bilinear estimate of Strichartz type and Bourgain's idea to split the data into low- and high-frequency parts.

This paper is concerned with the perturbed simple pendulum problem $$ -u''(t) + g(u(t)) = \lambda \sin u(t), \ \ u(t) > 0, \ \ t \in I := (-T, T), \ \ u(\pm T) = 0, $$ where $\lambda > 0$ is a parameter. It is known that if $\lambda \gg 1$, then the corresponding solution develops the boundary layers. We adopt a new parameter $\epsilon \in (0, T)$ which characterizes both the boundary layers and the height of the solution and parametrize a solution pair $(\lambda, u)$ by $\epsilon$, namely, $(\lambda, u) = (\lambda(\epsilon), u_\epsilon),$ and establish the three-term asymptotics for $\lambda(\epsilon)$ as $\epsilon \to 0$.

In this paper we study the solutions of some semilinear parabolic problems with nonconstant coefficients. We prove the existence of solutions which blow up at a finite time, and give the behavior near a point of blow-up.