Exponential decay of solutions to nonlinear elliptic equations with potentials

Abstract

Abstract. Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in R n , where the linear term is given by Schrödinger operators H = −∆ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V , some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V (x) = |x| 2 .

On decay of solutions to nonlinear Schrödinger equations

“…There are a number of results on exponential decay of solutions to equation (1) (see [1,3,4,10,11,15]). Most of them, except [10,11], deal with the case when 0 is below the essential spectrum of H. However, the case when 0 is in a spectral gap is extremely important for applications [10].…”

On a decay property of solutions to the Haraux–Weissler equation

“…We employ a similar method in [5] to prove a sharp exponential decay for solutions to a semilinear elliptic equation arising in the study of standing waves for nonlinear Schrödinger equations. We also refer to the proof by Cazenave [1] which has a relation with the method of proof in [5].…”

On Gaussian decay estimates of solutions to some linear elliptic equations and their applications

“…In [16] or [13] the maximum principle or the ODE methods are essentially used. On the other hand, Fukuizumi-Ozawa [9] discussed complex valued solutions when k = 1 and 1 < p < ( n+2 n−2 ) + , and established the estimate…”

“…We remark that in [9] more general potentials other than the harmonic potential |x| 2 in (1.38) are treated. In Pankov [20] nonlinear elliptic equations of the form −∆u + V (x)u = f (x, u), which includes (1.38), are discussed and some exponential upper bounds of solutions are obtained.…”

“…Indeed, in this case if a solution v to (1.38) belongs to X then it must belong to L ∞ (R n ); see [9] or [20]. This paper is organized as follows.…”