Nonlinear Analysis-Theory Methods & Applications最新文献

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish that the global entropy weak solutions, constructed in Amadori and Christoforou (2022) to the Cauchy problem for any $BV$ initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein, admit unconditional time-asymptotic flocking without any further assumptions on the initial data. In addition, we show that the convergence to a flocking profile occurs exponentially fast.

We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge–Ampère Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate which is from the Monge–Ampère eigenvalue problem. As a result, we obtain the sharp boundary estimates and the optimal global Hölder regularity for the two equations.

We prove Ambrosetti–Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order $s\in (0,1)$. We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,\alpha }$ under some regularity assumptions in the nonlinearities, that is, the solutions of the mentioned equations are classical. We finish the work obtaining Ambrosetti-Prodi-type results for a problem with singular nonlinearities.

In this work we study the existence and uniqueness of a positive, as well as a sign-changing steady-state solution of the degenerate logistic equation with a non-homogeneous superlinear term. Our outcome on a solution that changes sign, defined in higher dimensions, contribute to the existing literature of a few results for the problem, mostly developed in one dimension. We apply variational techniques, in particular the problem constrained to the Nehari manifold, and investigate how it changes as the parameter $\lambda$ in the equation or the function $b$ vary, affecting the existence and non-existence of solutions of the elliptic problem.

We give another proof of a theorem of Rabinowitz and Stredulinsky obtaining infinite transition solutions for an Allen–Cahn equation. Rabinowitz and Stredulinsky have constructed infinite transition solutions as locally minimal solutions, but it is still an interesting question to establish these solutions by other method. Our result may attract the interest of constructing solutions with the shape of locally minimal solutions of Rabinowitz and Stredulinsky for problems defined on descrete group.

The paper focuses on the $L^p$-Positivity Preservation property ($L^p$-PP for short) on a Riemannian manifold $(M,g)$. It states that any $L^p$ function $u$ with $1<p<+\infty$, which solves $(-\Delta +1)u\ge 0$ on $M$ in the sense of distributions must be non-negative. Our main result is that the $L^p$-PP holds if (the possibly incomplete) $M$ has a finite number of ends with respect to some compact domain, each of which is $q$-parabolic for some, possibly different, values $2p/(p-1)<q\le +\infty$. When $p=2$, since $\infty$-parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the $L^p$-PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than $2p/(p-1)$ or with a uniform Minkowski-type upper estimate of order $2p/(p-1)$. The threshold value $2p/(p-1)$ is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the $L^p$-PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis–Kato inequality, Liouville-type theorems in low regularity, removable singularities results for