Search

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR]

This paper aims to give an affirmative answer to a conjecture raised by Soave (2020) [25] and considers the qualitative properties of normalized solutions to Sobolev critical/subcritical Schrödinger equations with combined nonlinearities. Precisely, we establish the mass threshold a ¯ such that the mountain pass type normalized solution exists for the Sobolev critical/subcritical Schrödinger equation with combined mass critical and mass supercritical nonlinearities. We then show that a ¯ is also a threshold of the limit behavior of the mountain pass type normalized solution of the Schrödinger equation with combined nonlinearities as the exponent of lower order term tending to the mass critical exponent. Among which, the results in the case that the mass small than the threshold a ¯ give an affirmative answer to the conjecture raised by Soave. [ABSTRACT FROM AUTHOR]

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure. [ABSTRACT FROM AUTHOR]

We consider nonautonomous master equations of finite-state, continuous-time Markovian jump processes with uniformly continuous and bounded transition matrix functions. The Earnshaw–Keener conjecture states that if the omega-limit set of the transition matrix function contains at least one matrix which is neither decomposable nor splitting, then the difference of any two probability distribution solutions tends to zero at infinity. The conjecture has been confirmed under the additional assumption that the transition matrix function is almost-automorphic. In this paper, we prove the conjecture in its full generality. [ABSTRACT FROM AUTHOR]