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Register a new userOn the logistic equation for the fractional p-Laplacian
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We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.
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We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $pge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Holder regularity up to the boundary, that is, $u/d^sin C^alpha(overlineOmega)$ for some $alphain(0,1)$, $d$ being the distance from the boundary.
We consider the Dirichlet problem for the nonlinear $p(x)$-Laplacian equation. For axially symmetric domains we prove that, under suitable assumptions, there exist Mountain-pass solutions which exhibit partial symmetry. Furthermore, we show that Semi-stable or non-degenerate smooth solutions need to be radially symmetric in the ball.
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Delta)^s_p$ with $sin(0,1)$ and $pge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Omega)$ and in the weighted Holder space $C^0_s(overlineOmega)$, respectively, do coincide.
We propose two asymptotic expansions of the two interrelated integral-type averages, in the context of the fractional $infty$-Laplacian $Delta_infty^s$ for $sin (frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland-Caffarelli-Figalli, 2012]. Our expansions are parametrised by the radius of the removed singularity $epsilon$, and allow for the identification of $Delta_infty^sphi(x)$ as the $epsilon^{2s}$-order coefficient of the deviation of the $epsilon$-average from the value $phi(x)$, in the limit $epsilonto 0+$. The averages are well posed for functions $phi$ that are only Borel regular and bounded.
We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Delta_p)^s u=g(u) textrm{ in } mathbb{R}^N, uin W^{s,p}(mathbb{R}^N),$$ where $sin (0,1)$, $2 leq p < infty$, $sp leq N $, $2 leq N in mathbb{N}$ and $(-Delta_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.
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