A priori estimates for some elliptic equations involving the pp-Laplacian

We consider the Dirichlet problem for positive solutions of the equation $-\Delta_p (u) = f(u)$ in a convex, bounded, smooth domain $\Omega \subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions guarantying $L^{\infty}$ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\infty}$ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains

Sectional symmetry of solutions of elliptic systems in cylindrical domains

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in \cite{DaPaSys, DaGlPa1, Pa, PaWe}.Comment: arXiv admin note: text overlap with arXiv:1209.5581, arXiv:1206.392

Symmetry results for cooperative elliptic systems via linearization

In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in \RN, $N \geq 2$. More precisely we prove that solutions having Morse index $j \leq N$ are foliated Schwarz symmetric if the nonlinearity is convex and a full coupling condition is satisfied along the solution

Qualitative properties of solutions of m-Laplace systems

We prove regularity results for the solutions of the equation -Delta(m)u = h(x), such as summability properties of the second derivatives and summability properties of 1/vertical bar Du vertical bar. Analogous results were recently proved by the authors for the equation -Delta(m)u = f (u). These results allow us to extend to the case of systems of m-Laplace equations, some results recently proved by the authors for the case of a single equation. More precisely we consider the problem {-Delta(m1)(u) = f (v) u > 0 in Omega, u = 0 on theta Omega {-Delta(m2)(v) = g(u) v > 0 in Omega, v = 0 on theta Omega and we prove regularity properties of the solutions as well as qualitative properties of the solutions. Moreover we get a geometric characterization of the critical sets Z(u) equivalent to {x is an element of Omega vertical bar Du(x) = 0} and Z(v) equivalent to {x is an element of Omega vertical bar Dv(x) = 0}. In particular we prove that in convex and symmetric domains we have Z(u) = {0} - Z(v), assuming that 0 is the center of symmetry

Morse Index of Solutions of Nonlinear Elliptic Equations

Morse index of solutions of semilinear elliptic equations : definition, computation and application

Some nonexistence results for positive solutions of elliptic equations in unbounded domains

We prove some Liouville type theorems for positive solutions of semilinear elliptic equations in the whole space $\mathbb{R}^N$, $N\geq 3$, and in the half space $\mathbb{R}^N_{+}$ with different boundary conditions, using the technique based on the Kelvin transform and the Alexandrov-Serrin method of moving hyperplanes. In particular we get new nonexistence results for elliptic problems in half spaces satisfying mixed (Dirichlet-Neumann) boundary conditions

A few symmetry results for nonlinear elliptic PDE on noncompact manifolds

International audienc