Sublinear Case Research Articles

Compactons in a class of doubly sublinear Gardner equations

We introduce and study a class of doubly sublinear Gardner equations ut+F(u;n)x+u3x=0 where F(u;n)=u1+n−κ1+2nu1+2n, which for 0<n induce solitons and in the doubly sublinear cases wherein −1/2<n<0, bi-directional compactons propagating in either direction. Their emergence, evolution, chase and head-on interactions are studied.

Multiple weak solutions for pi(x)‐Kirchhoff‐type quasilinear elliptic systems

The existence of infinitely many weak solutions for a class of quasilinear elliptic ‐Kirchhoff system is investigated by the symmetric mountain pass theorem. Furthermore, for the case where , I introduce a positive quantity to find a mountain pass weak solution and to consider the sublinear case under large growth conditions at infinity and zero.

Existence and multiplicity of solutions for m-polyharmonic Kirchhoff problems without Ambrosetti–Rabinowitz conditions

In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic m-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, we introduce a positive quantity λ M similar to the first eigenvalue of the m-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, 2011) and Bae-Kim (Mathematical Methods in the Applied Sciences, 2020).

Finite time extinction for a critically damped Schrödinger equation with a sublinear nonlinearity

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schrödinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{{\varepsilon}+(|u|^2)^\alpha}u,$ with $a\in{\mathbb{C}},$ ${\varepsilon}\ge0,$ $2\alpha=(1-m)$ and $m\in[0,1).$ Here, we consider the sublinear case $0 < m < 1$ with a critical damped coefficient: $a\in{\mathbb{C}}$ is assumed to be in the set $ D(m)=\big\{z\in{\mathbb{C}}; \; {\mathrm{Im}}(z) > 0 \text{ and } 2\sqrt m{\mathrm{Im}}(z)=(1-m){\mathrm{Re}}(z)\big\} . $ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropriate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.

Concave-convex critical problems for the spectral fractional laplacian with mixed boundary conditions

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, $$\begin{aligned} \left\{ \begin{array}{lll} (-\varDelta )^su=\lambda u^q+u^{2_s^*-1}, &{} u>0 &{} \hbox { in } \varOmega , \\ u=0&{} &{} \hbox { on } \varSigma _{\mathcal {D}}, \\ \displaystyle \frac{\partial u}{\partial \nu }=0&{} &{} \hbox { on } \varSigma _{\mathcal {N}}, \end{array} \right. \end{aligned}$$with \(\varOmega \subset {\mathbb {R}}^N\), \(N>2s\), a smooth bounded domain, \(\frac{1}{2}<s<1\), \(0<q<2_s^*-1\), \(q\ne 1\), being \(2_s^*=\frac{2N}{N-2s}\) the critical fractional Sobolev exponent, \(\lambda >0\), \(\nu \) is the outwards normal to \(\partial \varOmega \); \(\varSigma _{{\mathcal {D}}}\), \(\varSigma _{{\mathcal {N}}}\) are smooth \((N-1)\)-dimensional submanifolds of \(\partial \varOmega \) such that \(\varSigma _{{\mathcal {D}}}\cup \varSigma _{{\mathcal {N}}}=\partial \varOmega \), \(\varSigma _{{\mathcal {D}}}\cap \varSigma _{{\mathcal {N}}}=\emptyset \), and \(\varSigma _{{\mathcal {D}}}\cap {\overline{\varSigma }}_{{\mathcal {N}}}=\varGamma \) is a smooth \((N-2)\)-dimensional submanifold of \(\partial \varOmega \). In particular, we will prove that, for the sublinear case \(0<q<1\), there exists at least two solutions for every \(0<\lambda <\varLambda \) for certain \(\varLambda \in {\mathbb {R}}\) while, for the superlinear case \(1<q<2_s^*-1\), we will prove that there exists at least one solution for every \(\lambda >0\). We will also prove that solutions are bounded.

On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $$-\Delta _p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$$ in a bounded domain $$\Omega \subset {\mathbb {R}}^N$$ , where $$1<q<p$$ , $$\lambda \in {\mathbb {R}}$$ , and a is a sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in $$\{x\in \Omega : a(x)>0\}$$ , when the parameter $$\lambda $$ lies in a neighborhood of the critical value $$\lambda ^* := \inf \left\{ \int _\Omega |\nabla u|^p \, dx/\int _\Omega |u|^p \, dx: u\in W_0^{1,p}(\Omega ) {\setminus } \{0\},\ \int _\Omega a|u|^q\,dx \ge 0\,\right\} $$ . Among main results, we show that if $$p>2q$$ and either $$\int _\Omega a\varphi _p^q\,dx=0$$ or $$\int _\Omega a\varphi _p^q\,dx>0$$ is sufficiently small, then such solutions do exist in a right neighborhood of $$\lambda ^*$$ . Here $$\varphi _p$$ is the first eigenfunction of the Dirichlet p-Laplacian in $$\Omega $$ . This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $$q>p$$ or in the sublinear case $$q<p=2$$ the nonexistence takes place for any $$\lambda \ge \lambda ^*$$ . Moreover, we prove that if $$p>2q$$ and $$\int _\Omega a\varphi _p^q\,dx>0$$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $$\lambda ^*$$ , two of which are strictly positive in $$\{x\in \Omega : a(x)>0\}$$ .

Regularity theory for fractional reaction–subdiffusion equation and application to inverse problem

In this paper, we analyze the regularity in time of solutions for nonlinear fractional reaction–subdiffusion equations (FrRSEs). By constituting regularity estimates of resolvent operator and employing fixed point theorems, we establish some results on the global existence and regularity in time of solutions to FrRSE in two different cases of nonlinear perturbations, namely, sublinear and superlinear cases. As an application of these results, several results on the existence and regularity of solutions in an inverse source problem governed by FrRSE with the additional measurements given at terminal time will be shown.

Positive solutions to the nonlinear reactor model by variational iteration method

We investigate the equation of the form (a(t)u^{prime }(t))^{prime }+Aa(t)u^{prime }(t)+f(t,u(t))=0 a.e. in (0, 1) with boundary conditions u^{prime }(0)=0,u^{prime }(1)+Au(1)=0, which is associated with the classical Markus and Amundson’s model. The main goal of this paper is to show that He’s variational iteration method appears to be very useful in deriving formulas of approximate solutions and gives better results that other methods. This fact will be illustrated by a few representative examples of BVPs of the aforementioned type. The numerical part is preceded by the theoretical results devoted to existence of positive solutions to this problem and their properties. We also discuss the dependence of the solutions on functional parameters. Our approach allows us to consider sublinear and superlinear cases in which f can be singular with respect to the first variable.

Multiple solutions for p(x)${p(x)}$‐curl systems with nonlinear boundary condition

AbstractIn this paper, we study the ‐curl problem of the type: where is a bounded simply connected domain with a C1, 1 boundary denoted by , , is a continuous function, , and is a Carathéodory function. We obtain the existence and multiplicity of solutions for a class of ‐curl systems in the absence of Ambrosetti–Rabinowitz condition under superlinear case. Besides, we also obtain infinitely many solutions under sublinear case.

Nonhomogeneous quasilinear elliptic problems: linear and sublinear cases

We are concerned with a class of second order quasilinear elliptic equations driven by a nonhomogeneous differential operator introduced by C.A. Stuart [22] and whose study is motivated by models in Nonlinear Optics. We establish sufficient conditions for the existence of at least one or two non-negative solutions. Our analysis considers the cases when the reaction has either a sublinear or a linear growth. In the sublinear case, we also prove a nonexistence property. The proofs combine energy estimates and variational methods.

Multiplicity of positive solutions for singular elliptic problems

AbstractWe consider the following nonlinear singular elliptic equation where , . Our main purpose is to prove the existence of a large number of positive solutions with the asymptotic decay as . We also investigate the rate of decay of . These results are based on the sub and supersolution method and cover both sublinear and superlinear cases of f.

Existence of Solutions to a Class of Nonlinear Arbitrary Order Differential Equations Subject to Integral Boundary Conditions

We investigate the existence of positive solutions for a class of fractional differential equations of arbitrary order δ&gt;2, subject to boundary conditions that include an integral operator of the fractional type. The consideration of this type of boundary conditions allows us to consider heterogeneity on the dependence specified by the restriction added to the equation as a relevant issue for applications. An existence result is obtained for the sublinear and superlinear case by using the Guo–Krasnosel’skii fixed point theorem through the definition of adequate conical shells that allow us to localize the solution. As additional tools in our procedure, we obtain the explicit expression of Green’s function associated to an auxiliary linear fractional boundary value problem, and we study some of its properties, such as the sign and some useful upper and lower estimates. Finally, an example is given to illustrate the results.

Nontrivial solutions for a second order periodic boundary value problem with the nonlinearity dependent on the derivative

In this paper, we study the existence of nontrivial solutions to the nonlinear differential equation with derivative term u′′(t)+a(t)u(t)=f(t,u(t),u′(t)),t∈[0,ω],u(0)=u(ω),u′(0)=u′(ω), where a: [0,ω]→R+(R+=[0,+∞)) is a continuous function with a(t)⁄≡0, and f: [0,ω]×R×R→R is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on the nonlinearity, by using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial solutions to the above periodic boundary value problem is established in C1[0,ω] for the sublinear case. Moreover, two examples are given to demonstrate the validity of our main results.

Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

The aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

The fractional p-Laplacian evolution equation in $${\mathbb {R}}^N$$ in the sublinear case

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter $$1<p<2$$ and fractional exponent $$s\in (0,1)$$ . Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $$L^q$$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case $$p>2$$ has been dealt with in a recent paper. We study here the “fast” regime $$1<p<2$$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass value M and any p in the subrange $$p_c=2N/(N+s)<p<2$$ , and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all $$L^q$$ spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for $$p_c<p<p_1$$ , where $$p_1$$ is a new critical number that we introduce, $$p_1\in (p_c,2)$$ . We extend this type of singular solutions to the “very fast range” $$1<p<p_c$$ . They represent examples of weak solutions having finite-time extinction in that lower p range. We briefly examine the situation in the limit case $$p=p_c$$ . Finally, we show that very singular solutions are related to fractional elliptic problems of the nonlinear eigenvalue type, of interest in their own right.

Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant.

Existence of positive solutions for p-Laplacian systems involving left and right fractional derivatives

PurposeThe paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel'skii fixed-point theorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.Design/methodology/approachFixed-point theorems.FindingsNo finding.Originality/valueThe obtained results are original.

Existence of Positive Solution to Schrödinger-type Semipositone Problems with Mixed Nonlinear Boundary Conditions

We studied the existence of a positive solution to Schrodinger-type semipositone problems with mixed nonlinear boundary conditions. By considering the cases when the reaction term with a parameter satisfies a superlinear and a sublinear growth condition at infinity, we established the existence of a positive solution for the large and small values of the parameter, respectively. The proofs are mainly based on the sub- and supersolution method for the sublinear case and the mountain pass lemma with $C^{1,\alpha}(\overline{\Omega})$-regularity for the superlinear case.

Enhancing fracture-network characterization and discrete-fracture-network simulation with high-resolution surveys using unmanned aerial vehicles

A workflow is presented that integrates unmanned aerial vehicle (UAV) imagery with discrete fracture network (DFN) geometric characterization and quantification of fluid flow. The DFN analysis allows for reliable characterization and reproduction of the most relevant features of fracture networks, including: identification of orientation sets and their characteristics (mean orientation, dispersion, and prior probability); scale invariance in distributions of fracture length and spatial location/clustering; and the distribution of aperture values used to compute network-scale equivalent permeability. A two-dimensional DFN-generation approach honors field data by explicitly reproducing observed multi-scale fracture clustering using a multiplicative cascade process and power law distribution of fracture length. The influence of aperture on network-scale equivalent permeability is investigated using comparisons between a sublinear aperture-to-length relationship and constant aperture. To assess the applicability of the developed methodology, DFN flow simulations are calibrated to pumping test data. Results suggest that even at small scales, UAV surveys capture the essential geometrical properties required for fluid flow characterization. Both the constant and sublinear aperture scaling approaches provide good matches to the pumping test results with only minimal calibration, indicating that the reproduced networks sufficiently capture the geometric and connectivity properties characteristic of the granitic rocks at the study site. The sublinear aperture scaling case honors the directions of dominant fractures that play a critical role in connecting fracture clusters and provides a realistic representation of network permeability.

Compressive Sensing in Wireless Powered Network: Regarding Transmission as Measurement

With large numbers of nodes applied in wireless powered network (WPN), it becomes a great challenge for the orthogonal multi-access (OMA) schemes to collect data. In this letter, a novel WPN based on compressive sensing (CS) is proposed, which utilizes the sparsity of data and regards transmission as measurement. It collects the coherent measurements of sensing data from the concurrent transmission of all nodes and then recovers the sensing data by CS reconstruction algorithm, which indeed accelerates data collection. Compared with the WPN based on OMA, the proposed WPN is always superior on delay and owns higher collecting rate with the conditions of linear and sub-linear sparsity case satisfied. Further, when the allocation strategy is adjustable, a joint optimization of time and power is proposed based on maximizing the collecting rate. Simulation results illustrate the superiority of the proposed WPN.