Fractional p-Laplacian Operator Research Articles

The Sign-Changing Solution for Fractional (p,q)-Laplacian Problems Involving Supercritical Exponent

In this article, we consider the following fractional (p,q)-Laplacian problem (−Δ)ps1u+(−Δ)qs2u+V(x)(|u|p−2u+|u|q−2u)=f(u)+λ|u|r−2u, where x∈RN, (−Δ)ps1 is the fractional p-Laplacian operator ((−Δ)qs2 is similar), 0<s1<s2<1<p<q<Ns2, qs2*=NqN−s2q, r≥qs2*, f is a C1 real function and V is a coercive function. By using variational methods, we prove that the above problem admits a sign-changing solution if λ>0 is small.

Multiplicity of solutions for a scalar field equation involving a fractional p-Laplacian with general nonlinearity

We prove the existence of infinitely many radially symmetric solutions to the problem where , , , and is the fractional p-Laplacian operator. We treat both the cases sp = N and sp<N. The nonlinearity g is a function of Berestycki–Lions type with critical exponential growth if sp = N and critical polynomial growth if sp<N. We also prove the existence of a ground state solution for the same problem.

Maximum and anti-maximum principle for the fractional p-Laplacian with indefinite weights

We establish some non-resonance results and maximum and anti-maximum principles for the following quasilinear non local problem involving the fractional p-Laplacian operator perturbed by an indefinite potential{(−Δp)su+V|u|p−2u=λm|u|p−2u+f in Ω,u=0 in RN﹨Ω. Here p∈(1,+∞), s∈(0,1), V and m are indefinite and sign-changing functions of Lr(Ω) with r>max⁡{N,Nsp}; f∈Lr(Ω) is such that f≥0 and f≢0 in the regular bounded domain Ω⊂RN.

Hölder Continuity and Boundedness Estimates for Nonlinear Fractional Equations in the Heisenberg Group

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator on the Heisenberg-Weyl group mathbb {H}^n. Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.

Radial symmetry of a relativistic Schrödinger tempered fractional p-Laplacian model with logarithmic nonlinearity

In this paper, by introducing a relativistic Schrödinger tempered fractional p-Laplacian operator (–Δ)p,λs,m, based on the relativistic Schrödinger operator (–Δ + m2)s and the tempered fractional Laplacian (Δ + λ)β/2, we consider a relativistic Schrödinger tempered fractional p-Laplacian model involving logarithmic nonlinearity. We first establish maximum principle and boundary estimate, which play a very crucial role in the later process. Then we obtain radial symmetry and monotonicity results by using the direct method of moving planes.

On the nonlinear eigenvalue problems involving the fractional p-Laplacian operator with singular weight

On the nonlinear eigenvalue problems involving the fractional p-Laplacian operator with singular weight

On fractional p-Laplacian type equations with general nonlinearities

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

Growing solutions of the fractional [formula omitted]-Laplacian equation in the Fast Diffusion range

We establish existence, uniqueness as well as quantitative estimates for solutions u(t,x) to the fractional nonlinear diffusion equation, ∂tu+Ls,p(u)=0, where Ls,p=(−Δ)ps is the standard fractional p-Laplacian operator. We work in the range of exponents 0<s<1 and 1<p<2, and in some sections we need sp<1. The equation is posed in the whole space x∈RN. We first obtain weighted global integral estimates that allow establishing the existence of solutions for a class of large data that is proved to be roughly optimal. We use the estimates to study the class of self-similar solutions of forward type, that we describe in detail when they exist. We also explain what happens when possible self-similar solutions do not exist. We establish the dichotomy positivity versus extinction for nonnegative solutions at any given time. We analyse the conditions for extinction in finite time.

Existence of solutions for critical fractional p-Laplacian equations with indefinite weights

This article concerns the critical fractional p-Laplacian equation with indefinite weights $$ (-\Delta_p)^su=\lambda g(x)|u|^{p-2}u+h(x)|u|^{p_s^*-2}u \quad \text{in }\mathbb{R}^N, $$ where \(0&lt;s&lt;1&lt;p&lt;\infty\), \(N&gt;sp\) and \(p_s^*=Np/(N-sp)\), the weight functions \(g\) may be indefinite, and \(h\) changes sign. Specifically, based on the results of asymptotic estimates for an extremal in the fractional Sobolev inequality and the discrete spectrum of fractional p-Laplacian operator, we establish an existence criterion for a nontrivial solution to this problem.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/11/abstr.html

Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

The set of p-harmonic functions in B 1 is total in C k (B̄ 1)

ABSTRACT Let , with , be the fractional p-Laplacian operator. We prove that the span of p-harmonic functions in is dense in .

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

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&lt;s&lt;1&lt;r&lt;p&lt;+∞, sp&lt;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.

Critical Concave Convex Ambrosetti–Prodi Type Problems for Fractional 𝑝-Laplacian

Abstract In this paper, we consider a class of critical concave convex Ambrosetti–Prodi type problems involving the fractional p-Laplacian operator. By applying the linking theorem and the mountain pass theorem as well, the interaction of the nonlinearities with the first eigenvalue of the fractional p-Laplacian will be used to prove existence of multiple solutions.

Properties of positive solutions to nonlocal problems with negative exponents in unbounded domains

We study nonlinear nonlocal problems involving the fractional p-Laplacian operator and nonlinearities with negative exponents in the whole space or in an unbounded epigraph domain. The symmetry and monotonicity results are proved by using a direct method of moving planes.

Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth

Abstract In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth: { ( - Δ p ) s ⁢ u = Q u ⁢ ( u , v ) + H u ⁢ ( u , v ) in ⁢ Ω , ( - Δ p ) s ⁢ v = Q v ⁢ ( u , v ) + H v ⁢ ( u , v ) in ⁢ Ω , u = v = 0 in ⁢ ℝ N ∖ Ω , u , v ≥ 0 , u , v ≠ 0 in ⁢ Ω , \left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&amp;\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&amp;&amp;\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&amp;\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&amp;&amp;% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&amp;\displaystyle\geq 0,\quad u,v\neq 0&amp;&amp;\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. where ( - Δ p ) s {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, p &gt; 1 {p&gt;1} , s ∈ ( 0 , 1 ) {s\in(0,1)} , p ⁢ s &lt; N {ps&lt;N} , p s * = N ⁢ p N - p ⁢ s {p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in ℝ N {\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with p &lt; q ≤ p s ∗ {p&lt;q\leq p^{\ast}_{s}} and Q u {Q_{u}} and Q v {Q_{v}} are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., s = 1 {s=1} ) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian ( - Δ ) s {(-\Delta)^{s}} (i.e., p = 2 {p=2} and 0 &lt; s &lt; 1 {0&lt;s&lt;1} ) has not been studied in the literature before.

Concentration of positive solutions for a class of fractionalp-Kirchhoff type equations

AbstractWe study the existence and concentration of positive solutions for the following class of fractionalp-Kirchhoff type problems:$$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) &amp; \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u&gt;0 &amp; \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$whereɛis a small positive parameter,aandbare positive constants,s∈ (0, 1) andp∈ (1, ∞) are such that$sp \in (\frac {3}{2}, 3)$,$(-\Delta )^{s}_{p}$is the fractionalp-Laplacian operator,f: ℝ → ℝ is a superlinear continuous function with subcritical growth andV: ℝ3→ ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potentialVattains its minimum values. Finally, we obtain an existence result whenf(u) =uq−1+ γur−1, where γ &gt; 0 is sufficiently small, and the powersqandrsatisfy 2p&lt;q&lt;p*s⩽r. The main results are obtained by using some appropriate variational arguments.

A limiting obstacle type problem for the inhomogeneous p-fractional Laplacian

In this manuscript we study an inhomogeneous obstacle type problem involving a fractional p-Laplacian type operator. First, we focus our attention in establishing existence and uniform estimates for any family of solutions $$\{u_p\}_{p \ge 2}$$ which depend on the data of the problem and universal parameters. Next, we analyze the asymptotic behavior of such a family as $$p \rightarrow \infty $$ . At this point, we prove that $$\displaystyle \lim \nolimits _{p\rightarrow \infty } u_p(x) = u_{\infty }(x)$$ there exists (up to a subsequence), verifies a limiting obstacle type problem in the viscosity sense, and it is an s-Holder continuous function. We also present several explicit examples, as well as further features of the limit solutions and their free boundaries. In order to establish our results we overcome several technical difficulties and develop new strategies, which were not present in the literature for this type of problems. Finally, we remark that our results are new even for problems governed by fractional p-Laplacian operator, as well as they extend the previous ones by dealing with more general non-local operators, source terms and boundary data.

Remarks on a nonlinear nonlocal operator in Orlicz spaces

Abstract We study integral operators L u ( χ ) = ∫ ℝ ℕ ψ ( u ( x ) − u ( y ) ) J ( x − y ) d y $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$ of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem L u = f $\mathcal{L}u=f$ in a bounded domain Ω , $\Omega ,$ and boundary condition u ≡ 0 on Ω c ; ${{\Omega }^{c}};$ both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem f ( u ) = λ ψ ( u ) . $f\left( u \right)=\lambda \psi \left( u \right).$

A multiplicity result for a class of fractional p-Laplacian equations with perturbations in ℝN

ABSTRACTThis paper deals with a class of nonlinear elliptic equations with perturbations in the whole space involving the fractional p-Laplacian. As a particular case, we investigate the following Schrödinger equations with perturbations: where is the fractional p-Laplacian operator, is a positive continuous function, is a perturbation. We first establish a compactness theorem which allows us to give some estimates of the energy levels where the Palais-Smale condition can fail. Furthermore, using Ekeland's variational principle and the mountain pass theorem, we obtain the existence of at least two distinct nonnegative weak solutions for the above-mentioned equations.

MULTIPLICITY RESULTS FOR A KIRCHHOFF SINGULAR PROBLEM INVOLVING THE FRACTIONAL P-LAPLACIAN

The aim of this paper is to study the multiplicity of solutions for a Kirchhoff singular problem involving the fractional p-Laplacian operator. Using the concentration compactness principle and Ekeland's variational principle, we obtain two positive weak solutions.