Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

Abstract

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities:where Ω ⊂ R n is a smooth bounded domain in R n containing 0 in its interior, and f, g ∈ C(Ω) with f + , g + ≡ 0 which may change sign in Ω. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for λ sufficiently small. The variational approach requires that 0 < α < 2, 0 < s < α < n, 1 < q < 2 < p ≤ 2 * α (s) := 2(n−s) n−α , and γ < γH (α), the latter being the best fractional Hardy constant on R n .

Multiple Solutions for a Fractional Laplacian System Involving Critical Sobolev-Hardy Exponents and Homogeneous Term

“…Proof. The proof is almost the same as in [18,Proposition 3.8] and the details are omitted. Now, we establish the existence of a local minimizer for I λ,µ on N + λ,µ .…”

“…Concerning the nonlocal problems with critical Sobolev-Hardy exponents, there has been little research up to now, see [18,20] and the references therein. In particular, Zhang and Hsu [20] concerned the following fractional elliptic system…”

“…Problems (1.2) and (1.3) aroused the interesting results due to the lack of compactness for involving the critical exponent; hence, the associated energy functionals do not satisfy the Palais-Smale condition in general. Moreover, the explicit formula of the ground states of limiting problem (2.9) is not clear, the standard variational argument is not applicable directly, which is the difficulty for the fractional Laplacian problem with Hardy potential and critical growth.Motivated by [20], in this paper we focus on the general case f , g possibly change sign in Ω and F positively 2 * s (β)-homogeneous, we shall complement the results of [18,20] and extend the results of [11,12,13,14] to the fractional Laplacian operator. Our main tool is the Nehari manifold methods which is…”

“…Motivated by [20], in this paper we focus on the general case f , g possibly change sign in Ω and F positively 2 * s (β)-homogeneous, we shall complement the results of [18,20] and extend the results of [11,12,13,14] to the fractional Laplacian operator. Our main tool is the Nehari manifold methods which is…”

Existence results for a fractional elliptic system with critical Sobolev‐Hardy exponents and concave‐convex nonlinearities

“…For the case and on , various studies concerning the existence and multiplicity of solutions for problem and have been presented in literature and references therein. With regard to , the existence of solutions for nonlinear elliptic problem with fractional Laplacian operator were studied extensively; the readers may refer to previous studies . For example, Zhang and Hsu concerned the problem when and obtained the existence of multiple positive solutions for this fractional Laplacian equation with concave‐convex nonlinearities.…”