Charles A. Swanson

The best sobolev constant

For an arbitrary positive integer m, N > 2m, and q = 2N/ (N - 2m), the smallest possible constant is obtained for the Sobolev embedding .Explicit radial functions which attain this constant are demonstrated.

Critical semilinear biharmonic equations in R N

An existence theorem is obtained for a fourth-order semilinear elliptic problem in R N involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L 2N/(N–4) (R N ) is attained by all translations and dilations of (1 + ∣x∣ 2 ) (4-N)/2 . The best constant is found to be

Positive solutions of semilinear elliptic problems in unbounded domains

Di= a/ax,, i= l,..., N; each a;,E C,,, ’ + YQ), b E Cg,,(Q), h(x) 3 b, > 0, 0 < CI < 1; and f(x, U) satisfies assumptions (f,)-(fs) below. In particular it is assumed that f(x, 0) = 0 for all x E 52, implying that the boundary value problem (1.1) always has the trivial solution. Our main purpose is to establish the existence of a positive solution of (1.1) throughout 1;2 in cases for which the nonlinearity in (1.1) is unbounded above, i.e., f(x, r)/t -+ +co as t + +co locally uniformly in Q. We treat the case of bounded nonlinearities elsewhere [20]. The main Theorem 4.5

Radial entire solutions of a class of quasilinear elliptic equations

will be examined under suitable conditions on the functions f: 8, x R -+ R and g: R, -+R+, where R, =(O, 00); R, = [0, co), and 2 is a real parameter. The capillarity equation and the equation of prescribed mean curvature are important special cases of (1.1) for which g(p) = (1 + P’)-“~ and fir, U) is suitably specialized [ 1, 2, 3, 4, 10, 11, 12, 131. An entire solution of (1.1) is defined to be a function u E C*(R”) satisfying (1.1) at every point XE RN. Our primary objective is to obtain sufficient conditions on f and g for (1.1) to have positive radial entire solutions of the following three types:

Decaying solutions of semilinear elliptic equations in R N

This paper is concerned with the existence and asymptotic behavior of positive solutions of semilinear elliptic problems of second order in

Global positive solutions of semilinear elliptic equations

{\bf R}^N

Radial entire solutions to even order semilinear elliptic equations in the plane

,

Decaying Entire Positive Solutions of Quasilinear Elliptic Equations

N \geqq 2

Criticalp-Laplacian problems in RN

. Positive solutions in

An Lq(RN)-theory of subcritical semilinear elliptic problems

{\bf R}^N