Positive solutions of some quasilinear singular second order equations

Abstract

In this paper we study the existence and uniqueness of positive solutions of boundary value problems for continuous semilinear perturbations, say / : [0, 1) x (0, oo) -* (0, oo), of a class of quasilinear operators which represent, for instance, the radial form of the Dirichlet problem on the unit ball of R* for the operators: p-Laplacian (1 < p < oo) and k-Hessian (1 < k < N). As a key feature, f(r, u) is possibly singular at r = 1 or u = 0. Our approach exploits fixed point arguments and the Shooting Method.2000 Mathematics subject classification: primary 35J25, 35J65.

Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights

“…[2,10,12,18,23,42], and for general p > 1 see [17,20,24,50]; while in unbounded domains and for p = 2 cf. [27,31,42,47], and for general p > 1 see [1,11,14,21,25,36,45]. There is a large literature on p-Laplacian equations in the entire R n , but the nonlinear structure, objectives and methods differ somehow from those presented here.…”

Existence of radial solutions for the $p$-Laplacian elliptic equations with weights

“…Some existence and non-existence results for radial ground states of special cases of (2) are given in [5] when f is continuous also at u = 0 and non-negative for u > 0 small. In the recent paper [9] some existence, non-existence and uniqueness results for radial ground states of some special cases of (2) are given when f > 0 everywhere in R + but singular at u = 0. For a more detailed discussion and comparison with our results we refer to the Remarks after Theorems 5 and 6 in Section 7.…”

Blow-up solutions for a $p$-Laplacian elliptic equation of logistic type with singular nonlinearity

“…More exactly, the results that we will use has the following statement: for w 1 , w 2 ∈ X given. In [17], it was proved the following result Proof of Lemma 4.2: It is easy to check that u 0 (x) := Ax −α , x > 0 is a solution of (1.11), where A > 0 is the unique solution of (1.12). In the sequel, we will show that u 0 is a maximal solution for (1.11).…”

“…The proof of Lemma 4.2 is based upon ideas found in [17]. Here, we are able to prove that the solutions of problem (1.11) are of the form u(x) = Ax −α , with A verifying (1.12), by using a result of [17] instead of the Poincaré-Bendixon's Theorem as used in [15]. More exactly, the results that we will use has the following statement: for w 1 , w 2 ∈ X given.…”

“…for w 1 , w 2 ∈ X given. In [17], it was proved the following result Lemma 5.1 Assume that w 1 , w 2 ∈ X, then…”