Qualitative properties of ground states for singular elliptic equations with weights

Abstract

In a series of papers the question of uniqueness of radial ground states of the equation ∆u + f(u) = 0 and of various related equations has been studied. It is remarkable that throughout this work (except in very special circumstances) nowhere is a spatially dependent term taken into consideration. Here we shall make a first attempt to study the uniqueness of ground states for such spatially dependent equations and to establish qualitative properties of solutions for this purpose. Mathematics Subject Classification (2000). Primary 35J70, Secondary 35J60

On a semilinear variational problem

“…We will now prove that decreasing solutions of (3.2) are compactly supported. This is a well-known fact for ground states of (3.2), which holds true under even more general non-linearities (see, e.g., [12]). We will give a short self-contained proof here.…”

“…2) and there are existence, uniqueness and symmetry results under various assumptions on the nonlinearity f for positive solutions of (1.2) decaying to 0 at infinity. We do not give a complete account on the literature here but just refer to [6,7,12] and the references therein, which will be of particular interest to our situation. Note, however, that in both examples discussed above, u in addition is a minimizer of I.…”

“…Having noted that all minimizers satisfy the same Euler-Lagrange equation, one could refer to uniqueness results as proved in [6,7,12]. However, an additional variational convexity argument will greatly simplify these approaches.…”

Dead Cores and Bursts for Quasilinear Singular Elliptic Equations

“…All these equations are discussed in detail in section 4 of [6], as special cases of the main example…”

“…In particular, in [6] conditions on the exponents were found so that, under appropriate behavior of the nonlinearity f , radial ground states for (6.2)-(6.4) are unique. We shall also be interested in the radial version of (6.1), when Ω is a ball B R centered at 0 with radius R > 0, namely, 1/p ds, r ≥ 0, (6.7)…”