We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension n ≥ 3. Using multiplier techniques in the style of Morawetz, we prove a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to cases with an absence of embedded eigenvalues and zero resonances, under explicit conditions on the potentials.

The spectral properties of a class of non-self-adjoint second-order elliptic operators with indefinite weight functions on unbounded domains Ω are investigated. It is shown, under an abstract regularity assumption, that the non-real spectrum of the associated elliptic operators in L2(Ω) is bounded. In the special case where Ω = ℝn decomposes into subdomains Ω+ and Ω− with smooth compact boundaries and the weight function is positive on Ω+ and negative on Ω−, it turns out that the non-real spectrum consists only of normal eigenvalues that can be characterized with a Dirichlet-to-Neumann map.

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave—convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

By variational methods, for a type of Yamabe problem with boundary, we construct multi-peak solutions whose maxima tend to the boundary as the parameter tends to 0+ under certain assumptions. We also obtain the asymptotic behaviours of the solutions.

A Stokesian fluid in motion along a porous medium saturated by the same fluid is modelled by the Beavers—Joseph—Saffman boundary-value problem to generalized Forchheimer—Stokes—Fourier systems: what we call the Beavers—Joseph—Saffman (BJS) problem. The model has nonlinear character given by the temperature dependence of physical parameters such as the viscosity, the permeability, the thermal conductivity and the thermal expansion. The paper is concerned with the study of the steady-state and the time-dependent regimes via the Galerkin and the Faedo—Galerkin techniques, respectively.

We consider the following equations involving negative exponent:

where p > 0. Under optimal conditions on the parameters α > −2 and p > 0, we prove the non-existence of finite Morse index solution on exterior domains or near the origin. We also prove an optimal regularity result for solutions with finite Morse index and isolated rupture at 0.

Recently it was shown that if D is a bounded domain in ℂ whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g ∈ A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on D which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in ℂ such that D is finitely connected and such that D is equal to the interior of D.

The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.

We analyse the multiplicity of non-negative solutions for the critical concave—convex-type equation

where Ω is a bounded domain of ℝN, 1 < r < p and λ > 0, and μ is a real parameter. Combining minimization on the underlying Nehari manifold with energy estimates, we show that, under suitable conditions on a and b, three non-negative solutions may exist when λ is positive and sufficiently small and μ is in a right neighbourhood of μ1, the first weighted eigenvalue of the p-Laplacian. To the best of our knowledge, our multiplicity result is new, even in the semi-linear case p = 2.

We obtain a solution, the L∞-norm of which is greater than ⅓√3, to the semilinear system arising in the mathematical theory of superconductivity when the domain is a disc or the exterior of a disc in ℝ2. We obtain an estimate of the critical field for the arbitrary penetration depth and determine the critical field as the penetration depth goes to zero when the domain is a disc. We prove that the L∞-norm of the symmetric solutions on the inner boundary is greater than that of the outer boundary when the domain Ω is an annulus. Stability of the solutions is also studied.

The period function of Liénard systems ẋ = −y + F(x), ẏ = g(x) associated with a centre is considered and sufficient conditions are presented for the period function to be strictly and globally monotone increasing and to have at least one critical point. Examples of applications and remarks on related work are also given.