An existence result for super-critical problems involving the fractional p-Laplacian in RN

doi:10.1016/j.aml.2022.108422

Applied Mathematics Letters

Volume 135, January 2023, 108422

https://doi.org/10.1016/j.aml.2022.108422 Get rights and content

Abstract

In this paper we establish an existence result for the following super-critical problems involving the fractional $p$ -Laplacian ${(- Δ)}_{p}^{s} u + V (x) {| u |}^{p - 2} u = {| u |}^{ς - 2} u + λ {| u |}^{q - 2} u, in R^{N},$ where $0 < s < 1$ , $1 < q < p < ς$ and $s p < N$ . Here the nonlinearity ${| u |}^{ς - 2} u$ imposes no any growth conditions. This is a new result for super-critical values of $ς$ .

Section snippets

Introduction and main result

In this paper, we consider a class of important non-local problem ${(- Δ)}_{p}^{s} u + V (x) {| u |}^{p - 2} u = {| u |}^{ς - 2} u + λ {| u |}^{q - 2} u, in R^{N},$ where $0 < s < 1$ , $1 < q < p < ς$ , $s p < N$ and $λ$ is a real parameter. On the potential function $V$ we require

$(V_{1}) V \in C (R^{N}, R)$ and $0 < V_{0} \leq V (x)$ for any $x \in R^{N}$ .

$(V_{2}) V^{- 1} \in L^{\frac{1}{p - 1}} (R^{N})$ .

Here ${(- Δ)}_{p}^{s}$ is the fractional $p$ -Laplace operator which may be defined along a function $u \in C_{0}^{\infty} (R^{N})$ as ${(- Δ)}_{p}^{s} u (x) = 2 lim_{ϵ \to 0^{+}} \int_{R^{N} ∖ B_{ϵ} (x)} \frac{{| u (x) - u (y) |}^{p - 2} (u (x) - u (y))}{{| x - y |}^{N + p s}} d y,$ for any $x \in R^{N}$ , where $B_{ϵ} (x) = {y \in R^{N} : | x - y | < ϵ}$ .

By variational methods,

Notations and preliminaries

Let $E$ be a reflexive Banach space and $E^{*}$ be topological dual of $E$ . The duality pairing between $E$ and $E^{*}$ is denoted by $〈 \cdot, \cdot 〉$ and defined by $〈 u, u^{*} 〉 = \int_{R^{N}} u (x) u^{*} (x) d x, \forall u \in E, u^{*} \in E^{*} .$ Suppose that $Ψ : E \to R \cup {+ \infty}$ is a proper, convex and lower semi-continuous map. Moreover, assume that $Ψ$ is G $\hat{a}$ teaux differentiable on $K$ which is a convex and weakly closed subset of $E$ and $D Ψ (u)$ denotes the G $\hat{a}$ teaux derivative of $Ψ$ at $u \in K$ . Define $Ψ_{K} (u) ≔ \{\begin{matrix} Ψ (u), & u \in K, \\ + \infty, & u \notin K . \end{matrix}$ Consider the functional $I_{K} : E \to R \cup {+ \infty}$ defined by $I_{K} (u) ≔ Ψ_{K} (u) - Φ (u),$

Proof of Theorem 1.1

To prove Theorem 1.1, we define $K = K (r) ≔ {u \in E \cap L^{\infty} (R^{N}) : {| u |}_{\infty} \leq r},$ for some $r > 0$ to be determined later. The following lemma shows that the set $K$ is weakly closed.

Lemma 3.1

Assume that $r > 0$ is fixed then $K (r)$ is weakly closed in $E$ .

Proof

Let ${u_{n}}$ be a sequence in $K (r)$ such that $u_{n} ⇀ u_{0}$ in $E$ . Clearly $u_{0} \in E$ because of the reflexivity of $E$ . Going if necessary to a subsequence, $u_{n} (x) \to u_{0} (x)$ a.e. $x \in R^{N}$ . This yields that $| u_{0} (x) | = {lim}_{n \to \infty} | u_{n} (x) | \leq r$ for a.e. $x \in R^{N}$ . Therefore, ${| u_{0} |}_{\infty} \leq r$ . □

Lemma 3.2

For all $u \in K (r)$ , one has ${| D Φ (u) |}_{\infty} \leq r^{ς - 1} + λ r^{q - 1} .$

Proof

By a

Acknowledgments

The authors are grateful to the anonymous referees for their useful comments and suggestions.

References (20)

AmbrosettiA. et al.
Combined effects of concave and convex nonlinearities in some elliptic problems
J. Funct. Anal.
(1994)
LaskinN.
Fractional quantum mechanics and Levy path integrals
Phys. Lett. A.
(2000)
AutuoriG. et al.
Elliptic problems involving the fractional Laplacian in $R^{N}$
J. Differential Equations
(2013)
MancinelliR. et al.
Front propagation in reactive systems with anomalous diffusion
Phys. D.
(2003)
FallM.M. et al.
Nonexistence results fora class of fractional elliptic boundary value problems
J. Funct. Anal.
(2012)
DavilaJ. et al.
Concentrating standing waves for the fractional nonlinear Schrödinger equation
J. Differential Equations
(2014)
CaffarelliL. et al.
Variational problems in free boundaries for the fractional Laplacian
J. Eur. Math. Soc.
(2010)
GarroniA. et al.
$Γ$ -Limit of a phase-field model of dislocations
SIAM J. Math. Anal.
(2005)
FelsingerM. et al.
The Dirichlet problem for nonlocal operators
Math. Z.
(2015)
FranzinaG. et al.
Fractional $p$ -eigenvalues
Riv. Math. Univ. Parma.
(2014)

There are more references available in the full text version of this article.

Cited by (0)

View full text

An existence result for super-critical problems involving the fractional p-Laplacian in RN

Abstract

Section snippets

Introduction and main result

Notations and preliminaries

Proof of Theorem 1.1

Acknowledgments

J. Funct. Anal.

Phys. Lett. A.

J. Differential Equations

Phys. D.

J. Funct. Anal.

J. Differential Equations

Variational problems in free boundaries for the fractional Laplacian

J. Eur. Math. Soc.

Γ-Limit of a phase-field model of dislocations

SIAM J. Math. Anal.

The Dirichlet problem for nonlocal operators

Math. Z.

Fractional p-eigenvalues

Riv. Math. Univ. Parma.

An existence result for super-critical problems involving the fractional $p$ -Laplacian in $R^{N}$

$Γ$ -Limit of a phase-field model of dislocations

Fractional $p$ -eigenvalues