Sublinear Nonlinearities Research Articles

Monotonicity in Half-spaces for p-Laplace Problems with a Sublinear Nonlinearity

Monotonicity in Half-spaces for p-Laplace Problems with a Sublinear Nonlinearity

Repeated S-shaped and Σ-shaped bifurcation curves for the one-dimensional non-autonomous Minkowski-curvature problem

We study the multiple existence results of positive solutions concerning one-dimensional non-autonomous Minkowski-curvature problems for certain ranges of a parameter. By providing a sufficient condition, we guarantee repeated S-shaped and repeated Σ-shaped global unbounded continua of positive solutions under superlinear and sublinear nonlinearities, respectively. This result is new even for autonomous cases.

New results concerning a singular biharmonic equations with p-Laplacian and Hardy potential

This paper deals with a singular biharmonic equations with p-Laplacian and Hardy potential. Using Mountain Pass Theorem and Fountain Theorem with Cerami condition, we obtain the existence and multiplicity of sign-changing high-energy solutions under some suitable conditions on the nonlinear term f ( x , u ) . In addition, by applying an abstract critical point theorem of Kajikiya in [A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370], a sequence of sign-changing solutions converging to zero for the biharmonic equations with sublinear nonlinearities is also obtained.

Probing Temperature-Induced Plasmonic Nonlinearity: Unveiling Opto-Thermal Effects on Light Absorption and Near-Field Enhancement.

Precise measurement and control of local heating in plasmonic nanostructures are vital for diverse nanophotonic devices. Despite significant efforts, challenges in understanding temperature-induced plasmonic nonlinearity persist, particularly in light absorption and near-field enhancement due to the absence of suitable measurement techniques. This study presents an approach allowing simultaneous measurements of light absorption and near-field enhancement through angle-resolved near-field scanning optical microscopy with iterative opto-thermal analysis. We revealed gold thin films exhibit sublinear nonlinearity in near-field enhancement due to nonlinear opto-thermal effects, while light absorption shows both sublinear and superlinear behaviors at varying thicknesses. These observations align with predictions from a simple harmonic oscillation model, in which changes in damping parameters affect light absorption and field enhancement differently. The sensitivity of our method was experimentally examined by measuring the opto-thermal responses of three-dimensional nanostructure arrays. Our findings have direct implications for advancing plasmonic applications, including photocatalysis, photovoltaics, photothermal effects, and surface-enhanced Raman spectroscopy.

On an anisotropic $ \overset{\rightarrow }{p}(\cdot) $-Laplace equation with variable singular and sublinear nonlinearities

<p>In the present paper, we study an anisotropic $ \overset{\rightarrow }{p}(\cdot) $-Laplace equation with combined effects of variable singular and sublinear nonlinearities. Using the Ekeland's variational principle and a constrained minimization, we show the existence of a positive solution for the case where the variable singularity $ \beta(x) $ assumes its values in the interval $ (1, \infty) $.</p>

Existence of infinitely many weak solutions to Kirchhoff–Schrödinger–Poisson systems and related models

In this article we investigate the existence of infinitely many weak solutions for Kirchhoff–Schrödinger–Poisson systems via the critical point theory. We obtain infinitely many large energy solutions and negative energy solutions of the system with superlinear nonlinearities by using the fountain theorem and the dual fountain theorem, respectively. Moreover, we establish the existence of infinitely many weak solutions of the problem with sublinear nonlinearities by applying the genus theory introduced by Krasnolsel’skii [Topological Methods in the Theory of Nonlinear Integral Equations (The Macmillan Company, New York, 1964)]. In particular, we do not use the classical Ambrosetti–Rabinowitz condition.

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.

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

A non‐local elliptic equation of Kirchhoff‐type for with Dirichlet boundary conditions is investigated for the cases where . It is well known that with the non‐local effect removed and , a branch of positive solutions bifurcates from infinity at and no positive solution exists whenever for some (see K. J. Brown, Calc. Var. 22, 483‐494, 2005),where is the principal eigenvalue of the linear problem . As a consequence of the non‐local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for . Moreover, regions with three positive solutions are found for small value of . Comparisons are also made of the results here with those of the elliptic problem in the absence of the non‐local term under the same prescribed conditions using numerical simulations.

PERIODIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS WITH PERTURBED AND SUB-LINEAR TERMS

In this paper, we study a class of perturbed discrete nonlinear Schr&#246;dinger equations with sub-linear nonlinearities at infinity and obtain the existence of solitons for this class of equations by using a generalized saddle point theorem. To the best of our knowledge, there is no published result focusing on this class of perturbed discrete nonlinear equations by this method.

Existence of a BV solution for a mean curvature equation

We prove the existence of a bounded variation solution for a quasilinear elliptic problem involving the mean curvature operator and a sublinear nonlinearity. We obtain such a solution as the limit of the solutions of another quasilinear elliptic problem involving a parameter p > 1 as p → 1 + . The analysis requires estimates independent on p , as well as a precise version of the weak Euler-Lagrange equation satisfied by the solution.

Multiple Nontrivial Solutions for a Nonlocal Problem with Sublinear Nonlinearity

In this paper, we study the following nonlocal problem − a − b ∫ Ω ∇ u 2 d x Δ u = λ u + f x u p − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where a , b &gt; 0 are constants, 1 &lt; p &lt; 2 , λ &gt; 0 , f ∈ L ∞ Ω is a positive function, and Ω is a smooth bounded domain in ℝ N with N ≥ 3 . By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided f ∞ is small enough.

Stability and interaction of compactons in the sublinear KdV equation

Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to symmetric compact perturbations with the same support is proven analytically. Dynamics of compactons is studied numerically, including evolution of pulse-like disturbances and interactions of compactons of the same or opposite polarities. Compactons interact inelastically, though almost restore their shapes after collisions. Compactons play a two-fold role of the long-living soliton-like structures and of the small-scale waves which spread the wave energy.

On a double phase problem with sublinear and superlinear nonlinearities

In this paper, we study a class of double-phase problems with sub-critical growth. Some new criteria to guarantee that the existence and multiplicity results for the considered problem is established by using variational methods and critical point theory.

Existence and multiplicity of solutions for Schrödinger equations with sublinear nonlinearities

<abstract> In the paper, we investigate a class of Schrödinger equations with sign-changing potentials $ V(x) $ and sublinear nonlinearities. We remove the coercive condition on $ V(x) $ usually required in the existing literature and also weaken the conditions on nonlinearities. By proving a Hardy-type inequality, extending the results in ^{[<xref ref-type="bibr" rid="b1">1</xref>]}, and using it together with variational methods, we get at least one or infinitely many small energy solutions for the problem. </abstract>

Existence and multiplicity results for Kirchhoff-type problem with sublinear nonlinearity

In this paper, we deal with the existence and multiplicity of solutions of Kirchhoff-type problem (0.1)−a+b∫RN|∇u|2dx△u+u=f(x,u),inRN,u∈H1(RN),where N≥3. Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrival solutions by using the genus properties in critical point theory.

Uniform decay estimates for solutions of a class of retarded integral inequalities

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities:y(t)≤E(t,τ)‖yτ‖+∫τtK1(t,s)‖ys‖ds+∫t∞K2(t,s)‖ys‖ds+ρ,t≥τ≥0.As a simple example of application, the retarded scalar functional differential equation x˙=−a(t)x+B(t,xt) is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system x˙=F0(t,x)+∑i=1mFi(t,x(t−ri(t))) on Rn with superlinear nonlinearities Fi (0≤i≤m). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions.The third example for application concerns the study of the dynamics of the functional cocycle system dudt+Au=F(θtp,ut) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous equilibrium solution Γ is obtained under the hyperbolicity assumption on operator A and some additional hypotheses, and the global asymptotic stability of Γ is also addressed.

On positive periodic solutions to second-order differential equations with a sub-linear non-linearity

The paper studies the existence and uniqueness of a positive periodic solution to the equation u′′=p(t)u−q(t,u), where p∈L([0,ω]) and q:[0,ω]×R→R is a Carathéodory function sub-linear in the second argument. The general results are applied to some particular cases such as the equation u′′=p(t)u−h(t)sinu with p,h∈L([0,ω]). This equation appears when approximating non-linearities in the equation of motion of a certain non-linear oscillator, namely, a pendulum deflected towards the two charged bodies.

Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments

We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of positive weak solution for classes of sublinear nonlinearities including logistic type. A method of sub- and supersolution, without monotone iteration, is established to prove our existence results. We also provide numerical bifurcation diagrams and the profile of positive solutions, corresponding to the theoretical results using the finite element method in one dimension.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/81/abstr.html

Remarks on infinitely many solutions for a class of Schrödinger equations with sublinear nonlinearity

In this paper, we study the existence of multiple nontrivial solutions for the following semilinear Schrödinger equation: where the potential V is continuous and allowed to be sign‐changing and f(x,u) satisfies more general sublinear growth conditions than those in previous studies. The first result of this paper obtains the existence of a nontrivial solution of (1). The second establishes an existent result on infinitely many nontrivial solutions of (1) by using a variant fountain theorems.

Periodic and Subharmonic Solutions for a Class of Sublinear First-Order Hamiltonian Systems

By the aids of variational methods, we study the existence of periodic and subharmonic solutions, and the minimality of their periods, for the nonautonomous first order Hamiltonian system $$J{\dot{u}}(t)+\nabla H(t,u(t))+h(t)=0$$ under some kind of sublinear nonlinearity conditions. Our results are new and generalize and extend some results in the literature.