In this paper we construct solutions which develop two negative spikes as $\varepsilon\to0^+$ for the problem $-\Delta u=|u|^{4/(N-2)}u+\varepsilon f(x)$ in $\varOmega$, $u=0$ on $\partial\varOmega$, where $\varOmega\subset\mathbb{R}^N$ is a bounded smooth domain exhibiting a small hole, with $f\geq0$, $f\not\equiv0$. This result extends a recent work of Clapp et al. in the sense that no symmetry assumptions on the domain are required.

We establish the persistence of the asymptotic stability of a linear equation $v'=A(t)v$ in a Banach space under sufficiently small perturbations, when the linear equation admits a non-uniform exponential contraction or a non-uniform exponential dichotomy. Moreover, we obtain optimal estimates for the decay of solutions of the perturbed equation, that in general may depend on the initial time.

Using Leray–Schauder degree theory, a theorem of upper and lower solutions and a strong maximum principle for the telegraph equation we prove an Ambrosetti–Prodi-type result for periodic solutions of the telegraph equation.

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 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.

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.

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.

We present a new necessary condition for similarity of indefinite Sturm–Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl–Titchmarsh $m$-functions. We also obtain necessary conditions for regularity of the critical points $0$ and $\infty$ of $J$-non-negative Sturm–Liouville operators. Using this result, we construct several examples of operators with the singular critical point $0$. In particular, it is shown that $0$ is a singular critical point of the operator

$$ -\frac{(\mathrm{sgn} x)}{(3|x|+1)^{-4/3}}\frac{\mathrm{d}^2}{\mathrm{d} x^2} $$

acting in the Hilbert space $L^2(\mathbb{R},(3|x|+1)^{-4/3}\,\mathrm{d} x)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a $J$-non-negative Sturm–Liouville operator of type $(\mathrm{sgn} x)(-\mathrm{d}^2/\mathrm{d} x^2+q(x))$ with the same properties.

We study an implicit second-order ordinary differential equation with complete integral. The class of second-order Clairaut-type equations is an important class of completely integrable equations. Indeed, the notion of second-order Clairaut-type equations is the generalization of second-order classical Clairaut equations. We give generic classifications of second-order Clairaut-type equations under the weak equivalence and the one-parameter strict equivalence relations.

We introduce the concept of causality into the framework of generalized pseudo-Riemannian geometry in the sense of Colombeau and establish the inverse Cauchy–Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward by Hawking and Ellis. Our work is based on a new characterization of free elements in finite-dimensional modules over the ring of generalized numbers.

A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

We prove weighted norm inequalities for vector-valued commutators of multilinear operators. We first consider the strong $(p,p)$ case with $p>0$ and then the case weak-type estimate.