Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up

Existence of Two Boundary Blow-Up Solutions for Semilinear Elliptic Equations

“…The growth condition F &1Â2 is necessary for the existence of the positive solution, as proved by Keller [10]. And the restriction to subcritical growth at & is needed, since the sign-changing solution may fail to exist for supercritical powers as shown by McKenna, Reichel, Walter [13] for f (s)= |s| p , p N* on balls and by our next theorem for other domains and nonlinearities. For simplicity, we state a weaker version of a more general theorem given in Section 8.…”

“…The first result of nonuniqueness was obtained by McKenna, Reichel, Walter [13], in the special case when the domain 0 is a ball and f (u)=|u| p . More precisely, they proved that for 1< p<N* (we write N*=(N+2)Â(N&2) for N 3 and N*= for N=1, 2), there are exactly two blow-up solutions: one positive and one sign-changing.…”

“…McKenna, Reichel and Walter prove in [13] the non-existence of the sign-changing blow-up solution for the special case f (s)=|s| p , p N* and 0=B R . For a slightly more general class of nonlinearities and some special symmetric domains including balls we have the following nonexistence result in the supercritical case.…”

Existences and Boundary Behavior of Boundary Blow-Up Solutions to Quasilinear Elliptic Systems With Singular Weights

“…Very recently, Zhang [33] and Yang [23] extended the above results to the problem (1.3) and gained some new results with nonlinear gradient terms. Problem (1.3) was discussed in a number of works; see, [2,3,4,5,9,10,11,12,13,19,23,25,34], Now let us return to problem (1.1). When m = n = 2, system (1.1) becomes 4) in the paper [14], when a(x) = 1, b(x) = 1, under Dirichlet boundary conditions of three different types: both components of (u, v) are bounded on ∂Ω (finite case); one of them is bounded while the other blows up(semilinear case); or both components blow up simultaneously(infinite case), under the assumption that(a − 1)(e − 1) > bc, necessary and suffcient conditions for existence of positive solutions were found, and uniqueness or multiplicity were also obtained, together with the exact boundary behavior of solutions.…”

Sturm-Liouville theory for the radial $\Delta_p$ -operator

“…If u(r 0 ) = 0 and u (r 0 ) > 0 or < 0, then ϕ(r 0 ) = 0 or π (mod 2π), resp. We write equation (4) in the form ϕ = f (r, ϕ) and note the following properties of f (r, s); it is assumed that…”