Classification of solutions to $Δu = u^{-γ}$ in the half-space
We provide a classification result for positive solutions to $Δu = u^{-γ}$ in the half space, under zero Dirichlet boundary condition.
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arXiv subjects
Berardino Sciunzi
Publications and source records attributed to Berardino Sciunzi.
Anisotropic Kelvin-type transform for a class of anisotropic elliptic operators
In this paper we introduce and study a new Kelvin-type transform in the anisotropic setting. In particular, we deal with semilinear and quasilinear anisotropic elliptic problems in the entire space.
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On the Pohozaev identity for quasilinear Finsler anisotropic equations
In this paper we derive the Pohozaev identity for quasilinear equations \begin{equation}\tag{$E$}\label{eq:p} -\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))=g(x, u) \quad \text {in}\,\, Ω, \end{equation} involving the anisotropic Finsler operator $-\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$. In particular, by means of fine regularity results on the vectorial field $B'(H(\nabla u))\nabla H(\nabla u)$, we prove the identity for weak solutions and in a direct way.
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Symmetry and monotonicity results for solutions of vectorial $p$-Stokes systems
In this paper we shall study qualitative properties of a $p$-Stokes type system, namely $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$},$$ where ${\boldsymbol Δ}_p$ is the $p$-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain $Ω$ and the function $\boldsymbol{ f}$, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to proceed with the moving-planes technique for systems. As far as we know, these are the first qualitative kind results involving vectorial operators.
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Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity
In this paper we prove the monotonicity of positive solutions to $ -Δ_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
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The moving plane method for doubly singular elliptic equations involving a first order term
In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov-Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.
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Regularity and symmetry results for nonlinear degenerate elliptic equations
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
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On the Gibbons' conjecture for equations involving the $p$-Laplacian
In this paper we prove the validity of Gibbons' conjecture for the quasilinear elliptic equation $ -Δ_p u = f(u) $ on $\mathbb{R}^N.$ The result holds true for $(2N+2)/(N+2) < p < 2$ and for a very general class of nonlinearity $f$.
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A weak comparison principle in tubular neighbourhoods of embedded manifolds
We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.
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A Liouville theorem for superlinear heat equations on Riemannian manifolds
We study the triviality of the solutions of weighted superlinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a Liouville--type theorem for solutions bounded from below with nonnegative initial data, under an integral growth condition on the weight.
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Hopf Lemma and regularity results for quasilinear anisotropic elliptic equations
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
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Radial symmetry for a quasilinear elliptic equation with a critical Sobolev growth and Hardy potential
We consider weak positive solutions to the critical $p$-Laplace equation with Hardy potential in $\mathbb R^N$ $$-Δ_p u -\fracγ{|x|^p} u^{p-1}=u^{p^*-1}$$ where $1<p<N$, $0\le γ<\left(\frac{N-p}{p}\right)^p$ and $p^*=\frac{Np}{N-p}$. The main result is to show that all the solutions in $\mathcal D^{1, p}(\mathbb R^N)$ are radial and radially decreasing about the origin.
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On the Höpf Boundary Lemma for quasilinear problems involving singular nonlinearities and applications
In this paper we consider positive solutions to quasilinear elliptic problem with singular nonlinearities. We provide a Höpf type boundary lemma via a suitable scaling argument that allows to deal with the lack of regularity of the solutions up to the boundary.
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Monotonicity and symmetry of singular solutions to quasilinear problems
We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure.
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A variational approach to nonlocal singular problems
We provide a suitable variational approach for a class of nonlocal problems involving the fractional laplacian and singular nonlinearities for which the standard techniques fail. As a corollary we deduce a characterization of the solutions.
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Qualitative properties of singular solutions to semilinear elliptic problems
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
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Monotonicity and rigidity of solutions to some elliptic systems with uniform limits
In this paper we prove the validity of Gibbons' conjecture for a coupled competing Gross-Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems.
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Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips
We consider nonnegative solutions to $-Δu=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity $f$. In fact we provide a unified approach that works in all the cases $f(0)<0$, $f(0)= 0$ or $f(0)> 0$. Furthermore we make the effort to deal with nonlinearities $f$ that may be not locally-Lipschitz continuous. We also provide explicite examples showing the sharpness of our assumptions on the nonlinear function $f$.
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