Bifurcation for Quasilinear Elliptic Singular BVP

For a continuous function g ≥ 0 on (0, + ∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ?u, ? Δu + g(u)|?u|², with a power type nonlinearity, λu ^p  + f ₀(x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.     

Dead cores of singular Dirichlet boundary value problems with ϕ-Laplacian

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable. This work was supported by grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Council of Czech Government MSM 6198959214.     

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ? ^N , N ≥ 2, possibly with boundary. Let Ω _R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω _R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δ_pu = f(u) in Ω, u = 0 on ?Ω, Ω ? R ^N a bounded smooth domain, is studied as ? → 0⁺, for a class of nonlinearities f(u) satisfying f(0) = f(z₁) = f(z₂) = 0 with 0 < z₁ < z₂, f < 0 in (0, z₁), f > 0 in (z₁, z₂) and f(u)/u^p–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0⁺. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Multiple solutions of sublinear Lane-Emden elliptic equations

We study the sublinear elliptic equation, −Δ u = |u|^psgn u + f(x,u) in the bounded domain Ω under the zero Dirichlet boundary condition. We suppose that 0 < p < 1 and |f(x,u)| is small enough near u = 0 and do not suppose that f(x,u) is odd on u. Then we prove that this problem has infinitely many solutions. Supported in part by the Grant-in-Aid for Scientific Research (C) (No. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

Existence of semilinear elliptic equations with prescribed limiting behaviour

In this paper, we consider semilinear elliptic equations of the form Δu + f(u) = 0 over a quarter space with Dirichlet boundary conditions. Given a suitable positive root z of f, we show how to construct a non‐negative bounded solution u converging to a one‐dimensional limiting profile V with V . This is established using Perron's method by constructing sub‐solutions and super‐solutions and employing a sliding argument. Copyright © 2016 John Wiley & Sons, Ltd.     

A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u _t = f(u)(Δu + a∫_Ω u(x, t)dx ? u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s ^p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.     

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.     

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u^? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.     

KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation:utt + △2u + σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.     

Existence of Solutions for Some Quasilinear Degenerate Elliptic Inclusions in Weighted Sobolev Spaces

We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form ?div(𝒜(x, u, ?u)) + f(x)g(u) ∈ H(x, u, ?u), where 𝒜(x, u, ?u) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H(x, u, ?u) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained.     

Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition

The existence of travelling wave solutions for the heat equation ∂_t u –Δu = 0 in an infinite cylinder subject to the nonlinear Neumann boundary condition (∂u /∂n) = f (u) is investigated. We show existence of nontrivial solutions for a large class of nonlinearities f. Additionally, the asymptotic behavior at ∞ is studied and regularity properties are established. We use a variational approach in exponentially weighted Sobolev spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A complete classification of bifurcation diagrams of a p -Laplacian Dirichlet problem

We study the bifurcation diagrams of (classical) positive solutions u with |u |_∞ ∈ (0, ∞) of the p -Laplacian Dirichlet problem (φ_p (u ′(x)))′ + λf_q (u (x))) = 0, –1 ≤ x ≤ 1, u (–1) = 0 = u (1), where p > 1, φ_p (y) = |y |^{p –2} y, (φ_p (u ′))′ is the one-dimensional p -Laplacian, λ > 0 is a bifurcation parameter, and the nonlinearity f_q (u) = |1 – u |^q is defined on [0, ∞) with constant q > 0. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane

In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations F(∇u,D²u)=f(u) in the half plane, where f is locally Lipschitz continuous (with f(0)?0) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the u or |∇u| are bounded.     

Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations

The existence and uniqueness of positive radial solutions of the equations of the type [IML0001] in B_R, p>1 with Dirichlet condition are proved for λ large enough and f satisfying a condition[IML0002] is non-decreasing on [IML0003] It is also proved that all the positive solutions in C₁ ⁰(B^R) of the above equations are radially symmetric solutions for f satisfying [IML0004] and λ large enough.     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m (u) = f(u) in a bounded smooth domain Ω, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator, a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results, together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z≡{x∈ Ω ∨ D(u)(x) = 0 = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u ∈ C²(Ω∖{0}). Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari.” Mathematics Subject Classification (1991) 35B05, 35B65, 35J70