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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

In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u = (|x|^{-\alpha}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-\Delta)_{A}^s$ is the fractional magnetic operator with $0 0$ is a positive parameter. The electric potential $V\in C(\mathbb{R}^N, \mathbb{R}^+_0)$ satisfies $V(x) = 0$ in some region of $\mathbb{R}^N$, which means that this is the critical frequency case. We first prove the $(PS)_c$ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term $M$ can vanish at zero.

/pdf/fractional-magnetic-schr-o-dinger-kirchhoff-problems-with-1f6b8jlu74.pdf

Fractional magnetic Schr\"{o}dinger-Kirchhoff problems with convolution and critical nonlinearities.

In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{*{20}c}
 {\overline {DA\left( {x,u\left( x \right),Du\left( x \right)} \right)} = B\left( {x,u\left( x \right),Du\left( x \right)} \right), x \in \Omega ,} \\
 {u\left( x \right) = 0, x\partial \Omega } \\

 \end{array}$$
 where D is a Dirac operator in Euclidean space, u(x) is defined in a bounded Lipschitz domain Ω in ℝ
 n
 and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space W
 0
 1,()
 (Ω,Cl
 n
 ) under appropriate assumptions.

/pdf/weak-solutions-for-elliptic-systems-with-variable-growth-in-3eovnzhlo9.pdf

Weak solutions for elliptic systems with variable growth in Clifford analysis

Multiple solutions for a noncooperative Kirchhoff‐type system involving the fractional p‐Laplacian and critical exponents

In this paper, we establish a Hodge-type decomposition of variable exponent Lebesgue spaces of Clifford-valued functions, where one of the subspaces is the space of all monogenic Lp(x)-functions. Using this decomposition, we obtain the existence and uniqueness of solutions to the homogeneous A-Dirac equations with variable growth under certain appropriate conditions and to the Stokes equations in the setting of variable exponent spaces of Clifford-valued functions.

Hodge decomposition of variable exponent spaces of Clifford-valued functions and applications to Dirac and Stokes equations

In this paper, we study the following coupled nonlocal system \begin{document}$ \begin{equation*} \begin{cases} (-\Delta)^{s}u-\lambda_{1}u = \mu_{1}|u|^{\alpha}u+\beta|u|^{\frac{\alpha-2}{2}}u|v|^{\frac{\alpha+2}{2}} & \text{in} \ \ \mathbb{R}^{N},\\ (-\Delta)^{s}v-\lambda_{2}v = \mu_{2}|v|^{\alpha}v+\beta|u|^{\frac{\alpha+2}{2}}|v|^{\frac{\alpha-2}{2}}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} $\end{document} satisfying the additional conditions \begin{document}$ \int_{\mathbb{R}^{N}}u^{2}dx = b^{2}_{1}\ \text{and} \ \int_{\mathbb{R}^{N}}v^{2}dx = b^{2}_{2}, $\end{document} where \begin{document}$ (-\Delta)^{s} $\end{document} is the fractional Laplacian, \begin{document}$ 0 , \begin{document}$ \mu_{1},\, \mu_{2}>0 $\end{document} , \begin{document}$ N>2s $\end{document} , and \begin{document}$ \frac{4s}{N} . We are concerned with the attractive case, which corresponds to \begin{document}$ \beta>0 $\end{document} . In the case of low perturbations of the coupling parameter, by using two-dimensional linking arguments, we show that there exists \begin{document}$ \beta_{1}>0 $\end{document} such that when \begin{document}$ 0 , then the system has a positive radial solution. Next, in the case of high perturbations of the coupling parameter, we prove that there exists \begin{document}$ \beta_{2}>0 $\end{document} such that the system has a mountain-pass type solution for all \begin{document}$ \beta>\beta_{2} $\end{document} . These results correspond to low and high perturbations with respect to the values of the coupling parameter \begin{document}$ \beta $\end{document} . This paper extends and complements the main results established in [ 2 ] for the particular case \begin{document}$ N = 3 $\end{document} , \begin{document}$ s = 1 $\end{document} , \begin{document}$ \alpha = 2 $\end{document} .

https://www.aimsciences.org/article/exportPdf?id=f09d4fd3-ab57-40b9-aeea-1205eaaf6dde

Binlin Zhang

Papers

Fractional magnetic Schr\"{o}dinger-Kirchhoff problems with convolution and critical nonlinearities.

Weak solutions for elliptic systems with variable growth in Clifford analysis

Multiple solutions for a noncooperative Kirchhoff‐type system involving the fractional p‐Laplacian and critical exponents

Hodge decomposition of variable exponent spaces of Clifford-valued functions and applications to Dirac and Stokes equations

Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case