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Methods Of Modern Mathematical Physics

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrodinger–Kirchhoff type $$\begin{aligned} M\left( \iint _{R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\varDelta )^s_pu+V(x)|u|^{p-2}u=f(x,u)+g(x) \end{aligned}$$
 in $${\mathbb {R}}^N$$
 , where $$(-\varDelta )^s_p$$
 is the fractional p-Laplacian operator, with $$0<s<1<p<\infty $$
 and $$ps<N$$
 , the nonlinearity $$f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$
 is a Caratheodory function and satisfies the Ambrosetti–Rabinowitz condition, $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+$$
 is a potential function and $$g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$
 is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p -Laplacian in $${\mathbb {R}}^N$$ R N

In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A ( x , ∇ u ) = λ | u | q ( x ) − 2 u in Ω u = 0 on ∂ Ω , where Ω is a bounded or an exterior domain of R N and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane–Emden equation and we focus on new phenomena which are due to the presence of variable exponents.

/pdf/nonlinear-elliptic-equations-with-variable-exponent-old-and-2il05tuszn.pdf

Nonlinear elliptic equations with variable exponent: Old and new

Abstract The purpose of this paper is mainly to investigate the existence of entire solutions of the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in ℝN. By using variational methods and topological degree theory, we prove multiplicity results depending on a real parameter λ and under suitable general integrability properties of the ratio between some powers of the weights. Finally, existence of infinitely many pair of entire solutions is obtained by genus theory. Last but not least, the paper covers a main feature of Kirchhoff problems which is the fact that the Kirchhoff function M can be zero at zero. The results of this paper are new even for the standard stationary Kirchhoff equation involving the Laplace operator.

Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

In this paper, we are interested in the multiplicity of solutions for a non-homogeneous p -Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex–concave nonlinearities: a + b ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p d x d y θ − 1 ( − Δ ) p s u = λ ω 1 ( x ) | u | q − 2 u + ω 2 ( x ) | u | r − 2 u + h ( x ) in R N , where ( − Δ ) p s is the fractional p -Laplace operator, a + b >0 with a , b ∈ R 0 + , λ>0 is a real parameter, 0 s 1 p ∞ with sp R N . Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle. A distinguished feature of this paper is that a may be zero, which means that the above-mentioned problem is degenerate. To the best of our knowledge, our results are new even in the Laplacian case.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2015.0034

Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities

In this paper, we study the multiplicity of solutions for an elliptic type problem driven by the variable-order fractional Laplace operator involving variable exponents. More precisely, we consider ( − Δ ) s ( ⋅ ) u + λ V ( x ) u = α | u | p ( x ) − 2 u + β | u | q ( x ) − 2 u in Ω , u = 0 in R N ∖ Ω , where N ≥ 1 , s ( ⋅ ) : R N × R N → ( 0 , 1 ) is a continuous function, Ω is a bounded domain in R N with N > 2 s ( x , y ) for all ( x , y ) ∈ Ω × Ω , ( − Δ ) s ( ⋅ ) is the variable-order fractional Laplace operator, λ > 0 is a parameter, V : Ω → [ 0 , ∞ ) is a continuous function, α , β > 0 are two parameters and p , q ∈ C ( Ω ) . Under some suitable assumptions, we show that the above problem admits at least two distinct solutions by applying the mountain pass theorem and Ekeland’s variational principle. Then we prove that these two solutions converge to two solutions of a limit problem as λ → ∞ . Moreover, we obtain the existence of infinitely many solutions for the limit problem.

Multiplicity results for variable-order fractional Laplacian equations with variable growth

The paper deals with the existence and multiplicity of solutions of the fractional Schrodinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{document}$\begin{equation*}	(a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\,	 \text{in $\mathbb{R}^N$},\end{equation*}$ \end{document} where \begin{document}$s∈ (0,1)$\end{document} , \begin{document}$N>2s$\end{document} , \begin{document}$a∈ \mathbb{R}^+_0$\end{document} , \begin{document}$b∈ \mathbb{R}^+_0$\end{document} , \begin{document}$θ∈[1,N/(N-2s))$\end{document} , \begin{document}$A:\mathbb{R}^N\to\mathbb{R}^N$\end{document} is a magnetic potential, \begin{document}$V:\mathbb{R}^N\to \mathbb{R}^+$\end{document} is an electric potential, \begin{document}$(-Δ )_A^s$\end{document} is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

/pdf/nonlocal-schrodinger-kirchhoff-equations-with-external-1v9ljceldm.pdf

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

Abstract This paper is aimed to study ground states for a class of fractional Schrödinger equations involving the critical exponents: ( - Δ ) α ⁢ u + u = λ ⁢ f ⁢ ( u ) + | u | 2 α * - 2 ⁢ u   in ⁢ ℝ N , $(-\\Delta)^{\\alpha}u+u=\\lambda f(u)+|u|^{2_{\\alpha}^{*}-2}u\\quad\\text{in }% \\mathbb{R}^{N},$ where λ is a real parameter, ( - Δ ) α ${(-\\Delta)^{\\alpha}}$ is the fractional Laplacian operator with 0 < α < 1 ${0<\\alpha<1}$ , 2 α * = 2 ⁢ N N - 2 ⁢ α ${2_{\\alpha}^{*}=\\frac{2N}{N-2\\alpha}}$ with 2 ≤ N ${2\\leq N}$ , f is a continuous subcritical nonlinearity without the Ambrosetti–Rabinowitz condition. Based on the principle of concentration compactness in the fractional Sobolev space and radially decreasing rearrangements, we obtain a nonnegative radially symmetric minimizer for a constrained minimization problem which has the least energy among all possible solutions for the above equations, i.e., a ground state solution.

https://www.degruyter.com/downloadpdf/j/anona.2016.5.issue-3/anona-2015-0133/anona-2015-0133.xml

Binlin Zhang

Papers

p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities

Multiplicity results for variable-order fractional Laplacian equations with variable growth

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

Ground states for fractional Schrödinger equations involving a critical nonlinearity