Semi-linear second-order elliptic equations in L1

Semi-linear Second-order Elliptic Equations in $l^{1}$

$\beta(x, u)$ is not monotone but has the same sign as $u$ , we prove the existence of solutions when $f(x)$ belongs to an Orlicz class arbitrarily close to $L^{1}(\Omega)$ . We also consider equation (1) with a nonlinear boundary condition. The linear case is considered by Stampacchia [18]. Our basic technique is to multiply the equation by various monotone functions of $u$ . This method was u...

Positive solutions to second order semi-linear elliptic equations

Here G ⊆ R (N ≥ 2) is an unbounded domain, and L is a second-order elliptic operator. We mainly confine ourselves to the cases F (x, u) = W (x)u with real p and W (x) a real valued function on G, and F (x, u) = g(u) with g : R→ R continuous and g(0) = 0. The operator L = H − V is of Schrödinger type, namely V = V (x) is a real potential and H = −∆ or more generally H = −∇ · a · ∇ is a second or...

Linear Elliptic Equations of Second Order

The classical Schauder type results on C-regularity of solutions are exposed for linear second order elliptic equations with Hölder coefficients. Our approach is based on equivalent seminorms in Hölder spaces C, which are similar to seminorms introduced by S. Campanato [1]. Under this approach, the C-estimates for solutions are derived from the maximum principle and the interior smoothness of h...

Linear Elliptic Equations of Second Order

3 4 CONTENTS Preface These lecture notes are intented as an introduction to linear second order elliptic partial differential equations. It can be considered as a continuation of a chapter on elliptic equations of the lecture notes [17] on partial differential equations. In [17] we focused our attention mainly on explicit solutions for standard problems for elliptic, parabolic and hyperbolic eq...

Linear Elliptic Equations of Second Order