We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model isHere Ω is an open bounded subset of R N (N ≥ 2), ∆ p u := div(|∇u| p−2 ∇u) (1 < p < N ) is the p-laplacian operator, µ is a nonnegative bounded Radon measure on Ω and H(s) is a continuous, positive and finite function outside the origin which grows at most as s −γ , with γ ≥ 0, near zero.2010 Mathematics Subject Classification. 35J60, 35J61, 35J75, 35R06.

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We study the asymptotic behavior, as γ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model iswhere Ω is an open, bounded subset of R N and f is a bounded function. We deal with the existence of a limit equation under two different assumptions on f : either strictly positive on every compactly contained subset of Ω or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to −∆v + |∇v| 2 v = f in Ω.

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ \left \{\begin {array}{ll} \displaystyle -{\Delta }_{p} u= \frac {f}{u^{\gamma }} + g u^{q} & \text { in } {\Omega }, \\ u = 0 & \text {on } \partial {\Omega }, \end {array}\right . $$ − Δ p u = f u γ + g u q in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N , Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.

We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.