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Nontrivial Solutions of Quasilinear Equations In BV

Marzocchi, Marco (1996)

Serdica Mathematical Journal

The existence of a nontrivial critical point is proved for a functional containing an area-type term. Techniques of nonsmooth critical point theory are applied.

Nontrivial solutions to boundary value problems for semilinear Δ γ -differential equations

Duong Trong Luyen (2021)

Applications of Mathematics

In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: - Δ γ u = f ( x , u ) in Ω , u = 0 on Ω , where Ω is a bounded domain with smooth boundary in N , Ω { x j = 0 } for some j , Δ γ is a subelliptic linear operator of the type Δ γ : = j = 1 N x j ( γ j 2 x j ) , x j : = x j , N 2 , where γ ( x ) = ( γ 1 ( x ) , γ 2 ( x ) , , γ N ( x ) ) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f ( x , ξ ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.

Nonuniqueness for some linear oblique derivative problems for elliptic equations

Gary M. Lieberman (1999)

Commentationes Mathematicae Universitatis Carolinae

It is well-known that the “standard” oblique derivative problem, Δ u = 0 in Ω , u / ν - u = 0 on Ω ( ν is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.

Norm inequalities for potential-type operators.

Sagun Chanillo, Jan-Olov Strömberg, Richard L. Wheeden (1987)

Revista Matemática Iberoamericana

The purpose of this paper is to derive norm inequalities for potentials of the formTf(x) = ∫(Rn) f(y)K(x,y)dy,     x ∈ Rn,when K is a Kernel which satisfies estimates like those that hold for the Green function associated with the degenerate elliptic equations studied in [3] and [4].

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