Narrow Results

This paper deals with semilinear elliptic problems of Dirichlet type, in star shaped domains. An abstract result is stated, which gives sufficient conditions for a positive solution of the problem to have strictly star shaped superlevels. Moreover, it turns out that the maximum point is the only critical point for these solutions.

Then we apply the result to the "single-peak" solutions of some widely studied problems. First a nonlinear and subcritical elliptic equation is considered, when the nonlinearity approaches the critical one. Then Schrödinger type problems are studied. Finally, the case when the potential is constant is also analyzed, on a bounded domain.

We study the geometry of complete generic Ricci solitons with the aid of some geometric-analytical tools extending techniques of the usual Riemannian setting.

We prove global Hölder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions à la Cabré.

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if $s\in \mathbb{N}$. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of $s$-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.

In this paper, we investigate various comparison principles for quasilinear elliptic equations of $p$-Laplace type with lower-order terms that depend on the solution and its gradient. More specifically, we study comparison principles for equations of the following form:

Our purpose is to generalize some recent comparison principles for operators driven by $p$-Laplacian to a wide class of quasilinear equations including $(p,q)$-Laplacian. It turns out, in particular, that adding a $q$-Laplacian to $p$-Laplacian allows to weaken the assumptions needed on the Hamiltonian of lower order terms. The results are specialized in the case that the Hamiltonian has at most polynomial growth in the gradient with coefficients depending on $x$ and $u$.

This paper deals with maximum principles for locally Lipschitz cost functionals, under constraints given by functional equations associated with pairs of closed range unbounded operators. The so called “range condition” introduced by the author, plays an essential role. Such abstract maximum principles have both a unifying effect in this area and applications to optimal control of some partial differential equations.

In this paper we consider the solvability of nonlinear parabolic differential equations with discontinuous nonlinearities, subjected to nonlocal conditions. We are concerned with the existence of solutions. Our technique is based on the Green's function for linear parabolic partial differential equations, the maximum principle, fixed point theorems for multivalued maps and the method of lower and upper solutions.