Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents

Abstract

In this paper, we study multiplicity of positive solutions for a class of Kirchhoff type of equations with the nonlinearity containing both singularity and critical exponents. We obtain two positive solutions via the variational and perturbation methods.

Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in $${\mathbb{R}^{3}}$$ R 3

“…This equation extends the classical D'Alembert wave equation for free vibrations of elastic strings, which takes into account the changes in length of the string produced by transverse vibrations; while purely longitudinal motions of a viscoelastic bar of uniform cross section and its generalizations can be found in [2][3][4][5][6][7][8]. A distinguishing feature of equation ( the equation is no longer a pointwise identity.…”

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity

“…on a smooth bounded domain Ω ⊂ R 3 and f : Ω×R → R a continuous function, has been extensively studied (see [1,3,10,11,[15][16][17]19,18,20,23,24,30]). Particularly, in [24] Sun and Tang have considered the following problem…”

“…In particular, when λ = 1, p = 5, problem (1.4) has at least two solutions for μ > 0 small enough, such as [2,6,12,25,[27][28][29]. However, the singular Kirchhoff type problems have few been considered, except for [15] and [17]. Liu and Sun in [17] have investigated problem (1.2) with f (x, u) = g(x)u −γ + λh(x) u p |x| s , and g, h ∈ C(Ω), 0 ≤ s < 1, 3 < p < 5 − 2s.…”

“…They proved that the non-degenerate case of problem (1.2) has at least two positive solutions for λ > 0 small enough by the Nehari manifold. More recently, Lei, Liao and Tang in [15] have considered the non-degenerate case of problem (1.2) with f (x, u) = u 5 + μu −γ , that is, problem (1.2) with singularity and critical exponents. By the variational methods, we obtained that problem (1.2) has at least two positive solutions for μ > 0 small enough.…”

Two positive solutions for quasilinear elliptic equations with singularity and critical exponents

“…We will remove the singularity by the perturbation method. Our idea comes from [24,27]. The energy functional corresponding to (1.1) is defined by…”