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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.

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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

In this paper, we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. Non-trivial solutions are obtained by computing the critical groups and Morse theory.
Our results extend some classical theorems for semilinear elliptic equations to the non-local fractional setting.

Existence of weak solutions for non-local fractional problems via Morse theory

In this paper, we use a variant of the mountain pass theorem to investigate the existence of nontrivial solutions for a degenerate Kirchhoff type problem driven by a nonlocal fractional -Laplace operator with homogeneous Dirichlet boundary conditions:where , is a smooth bounded domain of , is the fractional -Laplace operator with and , and is a Caratheodory function. The main feature of this paper is the fact that the coefficient of fractional Laplace operator can be zero at zero, that is, the problem is degenerate. The novelty here is that we may consider nonlinearities that satisfy a local -superlinear condition and may change sign.

Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition

This paper is concerned with the following singularly perturbed problem in H 1 ( R 2 ) − e 2 Δ u + V ( x ) u + A 0 ( u ( x ) ) u + ∑ j = 1 2 A j 2 ( u ( x ) ) u = f ( u ) , e ( ∂ 1 A 2 ( u ( x ) ) − ∂ 2 A 1 ( u ( x ) ) ) = − 1 2 u 2 , ∂ 1 A 1 ( u ( x ) ) + ∂ 2 A 2 ( u ( x ) ) = 0 , e Δ A 0 ( u ) = ∂ 1 ( A 2 | u | 2 ) − ∂ 2 ( A 1 | u | 2 ) , where e is a small parameter, V ∈ C ( R 2 , R ) and f ∈ C ( R , R ) . By using some new variational and analytic techniques joined with the manifold of Pohozaev–Nehari type, we prove that there exists a constant e 0 > 0 determined by V and f such that for e ∈ ( 0 , e 0 ] , the above problem admits a semiclassical ground state solution v ˆ e with exponential decay at infinity. We also establish a new concentration behaviour of { v ˆ e } as e → 0 . In particular, our results are available to the nonlinearity f ( u ) ∼ | u | s − 2 u for s ∈ ( 4 , 6 ] , which extend the existing results concerning the case f ( u ) ∼ | u | s − 2 u for s > 6 .

Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in H1(R2)

Ground state sign-changing solutions for the Schrödinger–Kirchhoff equation in R3

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1

https://www.aimsciences.org/article/exportPdf?id=2613e2a5-04e8-4440-9292-6b40e7dc22c7

Binlin Zhang

Papers

Existence of weak solutions for non-local fractional problems via Morse theory

Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition

Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in H1(R2)

Ground state sign-changing solutions for the Schrödinger–Kirchhoff equation in R3

A diffusion problem of Kirchhoff type involving the nonlocal fractional p -Laplacian