On inequalities between solutions of first order partial differential-functional equations
We study a family of semilinear reaction-diffusion equations on spatial domains , ε > 0, in lying close to a k-dimensional submanifold ℳ of . As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by . The definition of , given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...
The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator on is proved. In this note there is shown that in the cases , no other transforms of this kind exist and for case , all such transforms are described.
In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.
Interior -regularity for the gradient of a weak solution to nonlinear second order elliptic systems is investigated.
The aim of this paper is to show that the Liouville-type property is a sufficient and necessary condition for the regularity of weak solutions of quasilinear elliptic systems of higher orders.
Sufficient conditions are obtained so that a weak subsolution of , bounded from above on the parabolic boundary of the cylinder , turns out to be bounded from above in .