On rapidly oscillating solutions of a nonlinear elliptic equation

The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation ε 2△u = f(u). We describe the asymptotic behavior as ε tends to zero of the solutions on a spherical crown C of RN , (N ≥ 2) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics.Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.

Forcing the system by a drift

We establish apriori estimate for the solutions of a degenerate non-divergence nonlinear elliptic equation. For this goal we study forcing the system by a drift.

Topological degree methods for partial differential operators in generalized Sobolev spaces

The main aim of this paper is to prove, by using the topological degree methods, the existence of solutions for nonlinear elliptic equation Au = f where Au is partial dierential operators of general divergence form.

Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation- div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),where Ω is a bounded smooth domain of 𝕉N.

Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction

We consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.