We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x)     for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x)     for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0  for x ∈ G, ⎨z(x) = h(x)    for x ∈ ∂G ⎩ where $x=(x₁,...,x_m)\in G\subset {ℝ}^m$, G is an open and bounded domain with $C^{2+\alpha }$ (0 < α ≤ 1) boundary, the operator     Az := ∑j,k=1m ajk(x) (∂²z/(∂xj ∂xk)) is...

We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected $C$ ($\supset$-shape). Next, we show that there is a relationship between these two problems.

We study here an optimal control problem for a semilinear elliptic equation with an exponential nonlinearity, such that we cannot expect to have a solution of the state equation for any given control. We then have to speak of pairs (control, state). After having defined a suitable functional class in which we look for solutions, we prove existence of an optimal pair for a large class of cost functions using a non standard compactness argument. Then, we derive a first order optimality system assuming...

Let Ω ⊂ RN be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) up - b (x, t) uq in Ω × R, where 0 &lt; p, q &lt; 1 and λ &gt; 0. In some cases we also show the existence of solutions uλ in the interior of the positive cone and that uλ can...

We consider the problem$$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\&gt;0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \&gt;0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $div(x,\nabla u)+{a\left(x\right)\left|u\right|}^{p-2}u={g\left(x\right)\left|u\right|}^{p-2}u+h\left(x\right){\left|u\right|}^{s-1}u$ in ${ℝ}^N$ ⎨ ⎩ u > 0, $li{m}_{\left|x\right|\to \infty }u\left(x\right)=0$, where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.