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$\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*}$ |

$s∈ (0,1)$ |

$N>2s$ |

$a∈ \mathbb{R}^+_0$ |

$b∈ \mathbb{R}^+_0$ |

$θ∈[1,N/(N-2s))$ |

$A:\mathbb{R}^N\to\mathbb{R}^N$ |

$V:\mathbb{R}^N\to \mathbb{R}^+$ |

$(-Δ )_A^s$ |

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stress–strain relations are considered. Instead of a set of constant anisotropy coefficients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stress–strain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [

The paper is focused on existence of nontrivial solutions of a Schrödinger-Hardy system in the Heisenberg group, involving critical nonlinearities. Existence is obtained by an application of the mountain pass theorem and the Ekeland variational principle, but there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the Hardy terms as well as critical nonlinearities.

*globally proliferative*, see Theorem 3.3.

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{a-2}u+\lambda | u|^{q-2}u+h(|x|)f(u) = 0 $ and

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{p^*_\alpha-2}u+\lambda | u|^{q-2}u+h(|x|)f(u)= 0, $

where $1 < p < n$, $\alpha \in [0,p]$, $a \in [p,p^*_\alpha]$, $p_\alpha^*= p(n-\alpha)/(n-p)$, $\lambda, \mu \in R$ and $q \ge 1$, while $h: R^+ \to R^+_0$ and $f: R\to R$ are given continuous functions. The main tool for deriving nonexistence theorems for the equations is a Pohozaev--type identity. We first show that such identity holds true for weak solutions $u$ in $H^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the first equation and for weak solutions $u$ in $D^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the second equation. Then, under a suitable growth condition on $f$, we prove that every weak solution $u$ has the required regularity, so that the Pohozaev--type identity can be applied. From this identity we derive some nonexistence results, improving several theorems already appeared in the literature. In particular, we discuss the case when $h$ and $f$ are pure powers.

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 $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

*p*-Laplace elliptic equations with weights on domains of $\mathbb{R}^n$, which include several prototypes, and we show that there exist a dead core solution having a burst within the core. This result is obtained by using an existence theorem for ground states having compact support, proved in [4] by the authors, together with qualitative properties and an existence theorem for dead core solutions contained in a recent work of Pucci and Serrin, see [10].

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