Multi-parameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term

Abstract

We establish some bifurcation results for the boundary value problem −∆u = g(u)where Ω is a smooth bounded domain in R N , λ, µ ≥ 0, 0 < p ≤ 2, f is nondecreasing with respect to the second variable, and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u −α around the origin, for some 0 < α < 1. Our approach relies on finding explicit sub-and super-solutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u| p .

On a class of sublinear singular elliptic problems with convection term

“…We are seeking in this paper classical solutions of (1) λ , that is, solutions u ∈ C 2 (Ω) ∩ C(Ω) that verify (1) λ . Closely related to our problem is the following one, which has been considered in [15,16]:…”

The asymptotic behaviour of the unique solution for the singular Lane–Emden–Fowler equation

“…The problem arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically materials (see [10,13,17,25,29]). …”

“…The problem was discussed and extended to the more general problems in number of works; see, for instance, [2,[9][10][11][12][13][14][15][16][17][19][20][21]23,[28][29][30][31][32][33]. For k ≡ 1 on Ω, and g satisfying (g 1 ), Crandall et al [10,Theorems 2.2 and 2.7] showed that problem (1.1) has a unique solution u ∈ C 2+α (Ω) ∩ C(Ω).…”

On G-invariant solutions of a singular biharmonic elliptic system involving multiple critical exponents in R N $R^{N}$

“…Moreover, let us also mention that when μ = 0 and the right-hand side nonlinearity term |x| -s u 2 * (s)-1 in (1.2) is substituted by u q-1 with 1 < q ≤ 2 * , there have been a variety of remarkable results on G-invariant solutions in [9][10][11]. Furthermore, for other results about this aspect, see [12] with singular Lane-Emden-Fowler equations, [13] with singular p-Laplacian equations, [14] with biharmonic operators and [15] with p(x)-biharmonic operators [16], and monograph [17] with generalized Lane-Emden-Fowler equations or Gierer-Meinhardt systems involving singular nonlinearity. For the systems of singular elliptic equations involving critical exponents, a wide range of works concerning the solutions structures have been presented in recent years.…”