On some singular problems involving the fractional p(x,.) -Laplace operator

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

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Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

Open Mathematics ◽

10.1515/math-2020-0090 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1518-1530

Author(s):

Xuesen Qi ◽

Ximin Liu

Keyword(s):

Laplace Operator ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Second Fundamental Form ◽

Coefficient Function ◽

First Eigenvalue ◽

Initial Hypersurface ◽

The Mean ◽

The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0151 ◽

2020 ◽

Vol 10 (1) ◽

pp. 732-774

Author(s):

Zhipeng Yang ◽

Fukun Zhao

Keyword(s):

Small Parameter ◽

Variational Methods ◽

Critical Exponent ◽

Laplace Operator ◽

Sobolev Inequality ◽

Fractional Laplacian ◽

Singularly Perturbed ◽

Choquard Equation ◽

Fractional Choquard Equation ◽

Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

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