In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like $k^{-7/6}$ as $k\to0$. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function $k^{-7/6}$ that we obtained in an earlier paper.

This paper is contributed to the inhomogeneous semilinear elliptic equation

\begin{equation} \Delta u+K(|x|)u^{p}+f(x)=0\quad\text{in }\mathbb{R}^n, \tag{$*$} \label{*} \end{equation}

where

$$ \Delta=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}} $$

is the Laplacian operator, $n\geq3$, $p>1$, $f(x)\geq0$ and $K(|x|)>0$ is a given locally Hölder continuous function in $\mathbb{R}^n\setminus\{0\}$. The existence, non-existence and decay properties of positive solutions for \eqref{*} are obtained under some assumptions on $f(x)$ and $K(|x|)$ satisfying the slow-decay condition, i.e. $K(|x|)\geq C|x|^{l}$ at infinity for some constants $C>0$ and $l>-2$. The decay properties of positive solutions for $(\ast)$ are also discussed for the critical decay case on $K(|x|)$ with $l=-2$.

In this paper, assume that $q$ is a positive continuous function in $\mathbb{R}^{N}$ satisfying suitable conditions. We prove that the Dirichlet problem $-\Delta u+u=q(z)|u|^{p-2}u$ in an exterior domain admits at least two positive solutions and a solution which changes sign.

In this work we extend the results of Amar et al. to a family of abstract functionals of autonomous type satisfying suitable locality and additivity properties, and general integral growth conditions of superlinear type. We single out a condition which is necessary and sufficient in order for a functional of this class to admit an integral representation, and sufficient to have an integral representation for its lower-semicontinuous envelope. We also show that the integrand $F(x,q)$ satisfies some nice regularity properties in the $q$-variable, in particular a convexity-type property along lines. By adapting the reparametrization techniques introduced in an earlier work by the author to the case at issue, we then prove that the family of integral functionals associated to integrands of this kind do meet the condition mentioned above; in particular, it is closed by $\varGamma$-convergence.

We investigate critical points of the functional

over a ball in ℝn. Here, W is radially symmetric but not convex. We embed the functional into a family of functionals

where E0,0(u) = E(u). A global bifurcation analysis yields a branch of non-trivial critical points depending on λ and positive ε, where we can set λ = 0. The geometric properties preserved on that branch, due to the maximum principle, prove compactness such that the singular limit as ε ↘ 0 exists. Under natural conditions on W and G the critical point obtained in this way is a minimizer of the original functional. That plan can be carried out only under the restriction of radial symmetry, since the maximum principle applies only to special elliptic equations of fourth order. That restriction, however, is not essential since every minimizer of the functional is radially symmetric.

We investigate the behaviour of the eigenvalues of a self-adjoint Sturm–Liouville problem with a separated boundary condition when the interval of the problem shrinks to an end point. It is shown that all the eigenvalues, except possibly the first, approach $+\infty$. The choices of the boundary condition are found for which the first eigenvalue tends to $+\infty$, independent of the coefficient functions, and the same is done for the $-\infty$ limit. For the remaining choices of the boundary condition, several types of condition on the coefficient functions are given, so that the first eigenvalue has a finite or infinite limit and, when the limit is finite, an explicit expression for the limit is obtained. Moreover, numerous examples are presented to illustrate these results, and a construction is given to perturb the finite-limit case to the no-limit case.

We study the effect of the potential |y|α on the stability of entire solutions for elliptic equations on ℝN, N ≥ 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a by-product of our results, we will improve the known results about the regularity of such solutions.

The paper is concerned with a non-local time-delayed reaction–diffusion equation. We prove the (time) asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength, i.e. the so-called strong wavefront, by means of the (technical) weighted energy method, when the initial perturbation around the wave is small. The exponential convergent rate is also given. Selection of a suitable weight plays a key role in the proof.

We compare solutions of a class of degenerate parabolic equations on a Riemannian manifold $M$ with solutions of the equation on a model manifold. The class of equations under consideration contains both the parabolic $p$-Laplace equation and the porous medium equation. We prove that, under curvature conditions, solutions on model manifolds induce sub- or supersolutions on $M$. Using this result, we obtain curvature-dependent estimates for the speed of propagation of solutions.

The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix

$$ \begin{pmatrix} -\dfrac{\mathrm{d}}{\mathrm{d} x}\rho(x) \dfrac{\mathrm{d}}{\mathrm{d} x}+q(x) & \dfrac{\mathrm{d}}{\mathrm{d} x}\dfrac{\beta(x)}{x} \\[12pt] -\dfrac{\beta(x)}{x}\dfrac{\mathrm{d}}{\mathrm{d} x} & \dfrac{m(x)}{x^2} \end{pmatrix} $$

is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarantee that the operator is semi-bounded from below are derived. It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential operator is given by the range $\text{range}((m\rho-\beta^2)/\rho x^2)$ in the case where the quasi-regularity conditions are not satisfied.

We consider the following problem:

\begin{alignat*}{2} -\text{div}(A(x,u)\nabla u)&=u^s+f(x) & \quad &\text{in }\varOmega, \\ u(x)&\ge0 & & \text{in }\varOmega, \\ u(x)&=0 & & \text{on }\p\varOmega, \end{alignat*}

where $\varOmega$ is an open bounded subset of $\mathbb{R}^N$, $N\ge3$, and

$$ A:\varOmega\times\mathbb{R}\rightarrow M_{N\times N} $$

is an elliptic matrix such that when $u\to\infty$ is non-coercive.

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

Two non-trivial solutions for semilinear elliptic resonant problems are obtained via the Lyapunov—Schmidt reduction and the three-critical-points theorem. The difficulty that the variational functional does not satisfy the Palais—Smale condition is overcome by taking advantage of the reduction and a careful analysis of the reduced functional.

We study a parabolic population model in the full space and prove the global-in-time existence of a weak solution. This model consists of two strongly coupled diffusion equations describing the population densities of two competing species. The system features intrinsic growth, interspecies and intraspecies competition of the species, as well as diffusion, cross-diffusion and self-diffusion, and drift terms related to varying environment quality. The cross-diffusion terms can be large, making the system non-parabolic for large initial data. For the first time to our knowledge, solutions in unbounded domains are studied. The method of our proof is a combination of a time semi-discretization, a transformation of the system guaranteeing the positivity of the solution, a special entropy symmetrizing the system and compactness arguments.

This work is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form

$$ \mathcal{L}u+|u|^{q-1}u \leq\mathcal{L}v+|v|^{q-1}v, $$

where $q>0$ is a given real number and $\mathcal{L}$ is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form given by the relation

$$ \mathcal{L}=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\bigg[a_{ij}(x)\frac{\partial}{\partial x_j}\bigg]. $$

We assume that $n\geq2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots,n$, are measurable bounded functions on $\mathbb{R}^n$ such that $a_{ij}(x)=a_{ji}(x)$ and that the corresponding quadratic form is non-negative. The results obtained in this work were announced by the author in 2005.

By using the moving plane method combined with integral inequalities and Hardy's inequality, some new Liouville-type theorems for semilinear polyharmonic equations in $\mathbb{R}^N$ and in $\mathbb{R}^N_+$ are proved.

Let $H_n$, $n\ge3$, be the space of all $n\times n$ Hermitian matrices. Assume that a map $\phi:H_n\to H_n$ preserves commutativity in both directions (no linearity or bijectivity of $\phi$ is assumed). Then $\phi$ is a unitary similarity transformation composed with a locally polynomial map possibly composed by the transposition. The same result holds for injective continuous maps on $H_n$ preserving commutativity in one direction only. We give counter-examples showing that these two theorems cannot be improved or extended to the infinite-dimensional case.

A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary-value problems for many trial values of the spectral parameter. We present numerical results both for eigenvalues and for resonances, comparing with results reported by Aslanyan, Parnovski and Vassiliev.

A physically transparent and mathematically streamlined derivation is presented for a third-order nonlinear dynamical system that describes the curious chiral reversals of a celt (rattleback). The system is integrable, and its solutions are periodic, showing an infinite succession of spin reversals. Inclusion of linear dissipation allows any given number of reversals, and a typical celt's observed behaviour is well captured by tuning the dissipation parameters.

When a domain is inside a parabola-shaped region, we show that the principal eigenvalues for Dirichlet problems of a linear second-order elliptic operator in non-divergent form are bounded from below by a positive constant that is independent of the domain.