$\scriptstyle L^\infty$ is not dense in some function spaces like: the space $\scriptstyle EXP$ of exponentially integrable functions; the Marcinkiewicz space $\scriptstyle L^{q,\infty}=\text{weak-}L^q$; the Orlicz space $\scriptstyle L^A$ when the convex continuously increasing function $\scriptstyle A$, does not satisfy the so-called $\scriptstyle \Delta _2$-condition. We find formulas for the distance to $\scriptstyle L^{\infty}$ in these spaces. Using the simple observation that if a bounded linear operator $\scriptstyle T:L^q\to W$ satisfies $\scriptstyle T(L^{\infty})\subset L^{\infty}$, then $\scriptstyle \text{dist}_W(Tf,L^{\infty})=0,\quad \forall f\in L^q,$ we give some applications of previous results (see Section 5) to integrability properties of Riesz potential and of solutions to linear elliptic equations.

In this paper we develop an original variational approach which allows us to prove the existence of two positive solutions for a class of semilinear nonhomogeneous elliptic equations set on $\mathbb{R}^{N}$. In addition to a lack of compactness the main difficulty to overcome is the degenerated structure of the set of possible critical points.

We consider the following nonlinear multiparameter Sturm-Liouville problem: $$ \begin{align} u''(x) + \sum_{k=1}^n \mu_kf_k(u(x)) &= \lambda g(u(x)), \,\, u(x) > 0, \,\, x \in I := (0,1), \\ u(0) &= u(1) = 0, \end{align} $$ where $\mu = (\mu_1, \mu_2, \cdots, \mu_n) \in R_+^n \,\, (R_+ := (0, \infty))$ and $\lambda \in R_+$ are parameters. By using Ljusternik-Schnirelman theory on general level set due to Zeidler, the variational eigenvalues $\lambda = \lambda(\mu, \alpha)$ are obtained. Here, $\alpha > 0$ is a parameter of general level sets. We shall establish an asymptotic formula of $\lambda(\mu, \alpha)$ as $\mu_1 \to \infty$.

The role and importance of the Fourier method for investigation of mathematical physics problems is well known. By this method we can investigate problems for which the corresponding spectral problem is self-adjoint. By the Fourier method a solution $u(t,x)$ to the problem is written in the form of the series $u(t,x)=\sum^\infty_{k=1} C_k u_k(t,x),$ where $u_k(t,x)$ are elementary solutions of the considered problem. In the self-adjoint case the Hilbert theory of self-adjoint operators with a compact resolvent gives us information about the existence of elementary solutions and whether enough elementary solutions exist to write $u(t,x)$ in this series form. In the case when the principal part of the corresponding spectral problem is non-self-adjoint the existence of elementary solutions and convergence of the series are already problems. And this, in turn, leads to the problem of completeness of root functions. The completeness problem of the root vectors of a system of operator pencils and the root functions of elliptic boundary value problems is a subject of this paper.

We study variational integrals and related equations whose integrand grows almost linearly with respect to the gradient. A prototype of such functionals is $$ I[u] =\int |\nabla u|\,A\big(|\nabla u|\big)\,dx, $$ where $A$ is slowly increasing to $\infty $. For instance, $A(t)=\log^\alpha(1+t)$, $\alpha >0$, or $A(t)=\log\log (e+t)$, etc. We show that the minimizer $u$ subject to the Dirichlet data $v$ satisfies the estimate $$ \int |\nabla u|\,A^{1\pm\epsilon}\big(|\nabla u|\big)\,dx\le C\int|\nabla v|\, A^{1\pm\epsilon}\big(|\nabla v|\big)\,dx $$ at least for some small $\epsilon >0$. This extends previous results [12], [16] on integrals with power growth.

We give a necessary and sufficient condition for all solutions of a periodically forced Liénard system to be oscillatory. This paper also deals with the question when all trajectories keep on rotating around the origin. Several examples and global phase portraits are given to illustrate our results.

We consider a system of reaction-diffusion equations describing the dynamics of fission reactors with temperature feedback. There are $m$ equations for the neutrons in $m$ energy groups and a last temperature equation. We use the bifurcation method to find positive steady-states for the system which is not symmetric. We then analyze the linearized stability of the steady-state as a solution of the full system of $m+1$ parabolic equations. The asymptotic stability of the steady-state solution is proved by means of a stability theorem for sectorial operators.

We study the linear behavior of a weakly compressible viscoelastic fluid. Within an approximate theory of thermodynamics, compatible with the model, we construct the maximal free energy $\psi_M$ that is shown to induce a ``natural" norm $\Vert\cdot\Vert_M$ on the space state for such a fluid. Successively, since the stress $\mathbf{T}$ turns out to be continuous with respect to $\Vert\cdot\Vert_M$, we employ $\psi_M$ to obtain some stability and domain of dependence results.

We show well-posedness results for the generalized Korteweg-de Vries equation with nonlinear term $F(u)\partial_xu$. We assume $F(u)$ is a $C^4$ function and $F(0)=0$. Using a version of the chain rule for fractional derivatives and some estimates on the evolution group, we prove existence, uniqueness and regularity properties of the solution of the equation when the space of the initial data is $H^s(\mathbb{R}), \, s>1/2$. The theorem we prove is sharp. We obtain all the above results also for a mixed KdV and Schrödinger type equation proposed as a model for the propagation of a signal in an optic fiber.