Monotonicity results for the fractional p-Laplacian in unbounded domains

In this paper, we develop a direct method of moving planes in unbounded domains for the fractional p-Laplacians, and illustrate how this new method to work for the fractional p-Laplacians. We first proved a monotonicity result for nonlinear equations involving the fractional p-Laplacian in [Formula: see text] without any decay conditions at infinity. Second, we prove De Giorgi conjecture corresponding to the fractional p-Laplacian under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional p-Laplacian along a sequence of approximate maxima; (ii) estimating the lower bound of the solutions by constructing sub-solutions.

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

<p style='text-indent:20px;'>In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} &amp;u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &amp;u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ L_{\mathcal{A}} $\end{document}</tex-math></inline-formula> denotes a weakly <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>-coercive operator, which includes as prototype the <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-Laplacian or the generalized mean curvature operator, <inline-formula><tex-math id="M5">\begin{document}$ p,\,q&gt;0 $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M6">\begin{document}$ K\ast u^p $\end{document}</tex-math></inline-formula> stands for the standard convolution operator between a weight <inline-formula><tex-math id="M7">\begin{document}$ K&gt;0 $\end{document}</tex-math></inline-formula> satisfying suitable conditions at infinity and <inline-formula><tex-math id="M8">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>. For problem <inline-formula><tex-math id="M9">\begin{document}$ (P^-) $\end{document}</tex-math></inline-formula> we obtain a Fujita type exponent while for <inline-formula><tex-math id="M10">\begin{document}$ (P^+) $\end{document}</tex-math></inline-formula> we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.</p>

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>

On minimal decay at infinity of Hardy-weights

We study the behavior of Hardy-weights for a class of variational quasilinear elliptic operators of [Formula: see text]-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.

Harnack inequality for non-divergence structure semi-linear elliptic equations

In this paper we establish a Harnack inequality for non-negative solutions of {Lu=f(u)} where L is a non-divergence structure uniformly elliptic operator and f is a non-decreasing function that satisfies an appropriate growth conditions at infinity.