Sobolev versus Hölder local minimizers in degenerate Kirchhoff type problems

Abstract

In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type.Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is.We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering several different interactions between the nonlocal term and the nonlinearity.Mathematical Subject Classification MSC2010: 35J20 (35J70)

Existence of positive solutions for a class of semipositone problems with Kirchhoff operator

“…Proof. The proof was already given in [22,Proposition 4.1]. We give it here for sake of completeness.…”

“…For more recent literature about such Kirchhoff type problems we cite the works [2,27,16,15,8,13,32,31,5,4,30,21,28,22], which deal with the existence of solutions with various types of nonlinearities f and use mainly variational methods. Among them, we refer to [15,31,4,30,28,22] for considering also the case where the nonlocal term M is degenerate.…”

Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent

“…on the constraint S c := u ∈ H 1 (R 3 ) : ||u|| 2 2 = c 2 with Lagrange multipliers λ. We call 14 3 the L 2 -critical exponent for (1.1) λ , since inf u∈Sc E µ (u) > −∞ if q, p ∈ (2, 14 3 ) and inf u∈Sc E µ (u) = −∞ if 14 3 < q ≤ 6 or 14 3 < p ≤ 6. Taking a = 1 and b = 0, then (1.1) λ reduces to the classical Schrödinger equation:…”

“…) for free vibrations of elastic strings, where ρ denotes the mass density, u the lateral displacement, h the cross section area, ρ 0 the initial axial tension, E the Young modulus, L the length of the string and f the external force. In particular, (1.6) with M(0) = 0 models a string with zero initial tension, and is called the degenerate Kirchhoff equation, see [14,24]. One can refer to [1,6,8,12,13,20,9] and the references therein for more mathematical and physical background of (1.6).…”

“…In [32], H. Y. Ye studied (1.1) λ −(1.2) with a > 0, b > 0, µ = 0 and p ∈ (2,6). By considering a global minimization problem m(c, 0) := inf u∈Sc E 0 (u) > −∞, she proved that m(c, 0) is attained if and only if p ∈ (2, 10 3 ] and c > c * or p ∈ ( 10 3 , 14 3 ) and c ≥ c * , where c * :=    0, 2 < p < 10 3 ; a 3 4 W p 2 , p = 10 3 ; inf{c ∈ (0, +∞) : m(c, 0) < 0}, 10 3 < p < 14 3 , (see Lemma 2.2 below for W p ). When p = 14 3 , she showed that m(c, 0) has no minimizers for any c > 0.…”