Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

Abstract

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

Bifurcation-Type Results for the Fractional p-Laplacian with Parametric Nonlinear Reaction

“…Nonlocal elliptic equations driven by the fractional p-Laplacian with concave-convex reactions are investigated, for instance, in [2][3][4][5][6]. Other existence and bifurcation results for problems with several parametric reactions can be found in [7][8][9][10].…”

“…We will now establish some properties of λ * : Lemma 2. Let λ * be defined by Equation (5). Then, we have…”

“…In the next result, we deal with the threshold case λ = λ * : Lemma 3. Let λ * > 0 be defined by Equation (5). Then, (P λ * ) has at least one solution u * ∈ int(C 0 s (Ω) + ).…”

“…Proof. Let λ * > 0 be defined by (5). By Lemmas 2 and 4, for all λ ∈ (0, λ * ), problem (P λ ) has at least two solutions u λ , v λ ∈ int(C 0 s (Ω) + ) s.t.…”

Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN

“…However, this condition is too restrictive and gets rid of many nonlinearities. Thus many researchers have tried to drop the (AR)-condition in the elliptic problem of nonlocal type (see e.g., [20,[39][40][41][42]). In this respect, we are to prove the existence of a nontrivial solution for problem (3) without (AR)-condition using the mountain pass theorem with Cerami condition under a suitable assumption of the nonlinearity of f .…”