Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

Abstract

Abstract.Let Ω be a bounded domain in R n (n ≥ 2) with a smooth boundary ∂Ω. We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic systemHere r, s ∈ R, α, β < 1 such that γ := (1 − α)(1 − β) − rs > 0 and the functions ai (i = 1, 2) are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory.

Ground state solutions for a class of generalized quasilinear Schrödinger–Poisson systems

“…The semilinear case corresponding to ω = 0 has been studied extensively by many scholars in recent years (see [1][2][3][4][5]). Quasilinear equations of the form (1.2) have been derived as models of several physical phenomena corresponding to various types of l(s).…”

“…Hence the problem (1.4) can be rewritten in the following form: 5) whose corresponding variational functional is given by…”

On a class of stationary loops on SO ( n ) $\mathbf{SO}(n)$ and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint

“…Here, the first term on the right-hand side in the above is given by Now, moving forward by using the formulation (2.7), it is seen upon tensorisation that we have 9) where in deducing this identity use has been made of the pointwise relation…”

“…For the sake of clarity, note that by a [classical] solution we hereafter mean a pair (u, P) with u of class C 2 (U, R n ) ∩ C (U, R n ) and P of class C 1 (U) ∩ C (U) such that (1.2) holds in a pointwise sense in U. a Now, proceeding forward and arguing either formally and in a distributional sense, or classically, upon assuming further differentiability on L F , it is seen from (1.2)- Note, however, that this condition alone, unless U has a particular homology, does not imply that L F [u] is a gradient field in U, here, ∇P. For more on the background formulation and applications of system (1.2)-(1.3), in particular to function theory, mechanics, and nonlinear elasticity, see [2,3,5,10,14,19] and [1,4,7,11,12,[15][16][17][18] as well as [20][21][22][23][24][25][26][27]30] and [9,13,29,31,32] for related results and further applications.…”