On the Asymptotic Solution of a Partial Differential Equation with an Exponential Nonlinearity

Singular limits in Liouville-type equations

“…This second, "large" solution of (1.1) was found in simply connected domains in [32] when k ≡ 1, see also [11] for earlier work on existence. While subcritical in the sense of Trudinger-Moser embedding, this problem exhibits loss of compactness as ε → 0, similar to that present in equations at the critical exponent in higher dimensions.…”

Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term

“…His result has been generalized in [8] to (6) with γ < 1 4 , and finally by Ye in [9] to any exponentially dominated nonlinearity f(u). The existence of nontrivial branches of solutions with single singularity was first proved by Weston [10] and then a general result has been obtained by Baraket and Pacard [11]. These results were also extended, applying to the Chern-Simons vortex theory in mind, by Esposito et al [12] and Del Pino et al [13] to handle equations of the form -Δu = r 2 V(x)e u where V is a nonconstant positive potential.…”

On the nonlinear eigenvalue problem Δ𝑢+𝜆𝑒^{𝑢}=0

“…On the other hand, by the method of Weston [22], we get ||(1 -KJ-'Wco^c^D) < CX-1 as A I 0 except for a "pathological case" of n. Therefore, for n > 3 the iteration (2.6) converges to a genuine solution U* such that Here, Cy, C2 and C3 are positive constants. By the method of Wente [21], it can be shown that the "pathological case" does not arise when a = |a^(0)/rr^(0)| < 2.…”

“…Recently, under certain assumptions for n, V. H. Weston and J. L. Moseley have constructed a branch W* differing from W_ by the method of singular perturbations [22,16]. In the parametrization W* = {(At,wt)|2 < t < 3}, we have…”