Existence of positive solutions for nonlocal problems with indefinite nonlinearity

Abstract

In this paper, we consider the following new nonlocal problem:-(a-b Ω |∇u| 2 dx) u = λf (x)|u| p-2 u, x ∈ Ω, u = 0, x ∈ ∂Ω, where Ω is a smooth bounded domain in R 3 , a, b > 0 are constants, 3 < p < 6, and the parameter λ > 0. Under some assumptions on the sign-changing function f , we obtain the existence of positive solutions via variational methods.

New multiplicity of positive solutions for some class of nonlocal problems

“…For f (x, u) = λ|u| -γ with 0 < γ < 1, [6] got the multiplicity of positive solutions to (1.3). In [10], we proved that problem (1.3) possesses at least one positive solution when N = 3, f (x, u) = λf (x)|u| p-2 u with 3 < p < 4 and f (x) ∈ L 6 6-p ( ) may change sign. In particular, Duan et al [3] and Lei et al [7] proved that there exists λ * > 0 such that, for each λ ∈ (0, λ * ), problem (1.1) has two positive solutions by using the minimization argument and the mountain pass theorem.…”

Negative Energy Solutions for a New Fractionalpx-Kirchhoff Problem without the (AR) Condition

“…Our results generalize the related work in two ways. Firstly, we deal with the problem ðP λ Þ in the fractional framework and [32] consider only the integer framework. Secondly, compare to [33], we add a new Kirchhoff function and consider the case of variable exponents.…”

Multiple Nontrivial Solutions for a Nonlocal Problem with Sublinear Nonlinearity

“…However, we now face a new nonlocal term a − b Ð Ω j∇uj 2 dx, which is different from the well-known Kirchhoff type nonlocal term a + b Ð Ω j∇uj 2 dx. Now, there has been some results on the existence and multiplicity of nontrivial solutions to this new nonlocal problem (see [15][16][17][18][19][20][21][22]).…”