Showing papers in "Advances in Nonlinear Analysis in 2016"

Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations

Abstract: 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.

Fractional elliptic equations with critical exponential nonlinearity

Abstract: Abstract We study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

Fractional modelling arising in unidirectional propagation of long waves in dispersive media

Abstract: Abstract The purpose of this paper is to propose a modified and simple algorithm for fractional modelling arising in unidirectional propagation of long wave in dispersive media by using the fractional homotopy analysis transform method (FHATM). This modified method is an innovative adjustment in the Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. This modified method yields an analytical and approximate solution in terms of a rapidly convergent series with easily computable terms. The numerical solutions obtained by the proposed algorithm indicate that the approach is easy to implement and computationally very attractive. Comparing our solution with the existing ones, we note an excellent agreement.

Ground states for fractional Schrödinger equations involving a critical nonlinearity

Abstract: Abstract This paper is aimed to study ground states for a class of fractional Schrödinger equations involving the critical exponents: ( - Δ ) α ⁢ u + u = λ ⁢ f ⁢ ( u ) + | u | 2 α * - 2 ⁢ u in ⁢ ℝ N , $(-\\Delta)^{\\alpha}u+u=\\lambda f(u)+|u|^{2_{\\alpha}^{*}-2}u\\quad\\text{in }% \\mathbb{R}^{N},$ where λ is a real parameter, ( - Δ ) α ${(-\\Delta)^{\\alpha}}$ is the fractional Laplacian operator with 0 < α < 1 ${0<\\alpha<1}$ , 2 α * = 2 ⁢ N N - 2 ⁢ α ${2_{\\alpha}^{*}=\\frac{2N}{N-2\\alpha}}$ with 2 ≤ N ${2\\leq N}$ , f is a continuous subcritical nonlinearity without the Ambrosetti–Rabinowitz condition. Based on the principle of concentration compactness in the fractional Sobolev space and radially decreasing rearrangements, we obtain a nonnegative radially symmetric minimizer for a constrained minimization problem which has the least energy among all possible solutions for the above equations, i.e., a ground state solution.

Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

Abstract: We study the following class of nonlinear Choquard equation, $$ -\Delta u +V(x)u =\Big( \frac{1}{|x|^\mu}\ast F(u)\Big)f(u) \quad \mbox{in} \quad \R^N, $$ where $0<\mu

On the removability of isolated singular points for elliptic equations involving variable exponent

Abstract: Abstract In this paper, we study the problem of removable isolated singularities for elliptic equations with variable exponents. We give a sufficient condition for removability of the isolated singular point for the equations in W 1,p(x) (Ω)${W^{1,p(x)}(\\Omega )}$ .

On the Fractional p-laplacian equations with weight and general datum

Abstract: The aim of this paper is to treat the following problem $$ (P) \left\{ \begin{array}{rcll} (-\Delta)^s_{p, \beta} u &= & f(x,u) &\mbox{ in }\Omega, u & = & 0 &\mbox{ in } \mathds{R}^N\setminus\Omega, \end{array} \right. $$ where $$ (-\Delta)^s_{p,\beta}\, u(x):=P.V. \int_{\mathds{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}} \frac{dy}{|x|^\beta|y|^\beta},$$ $\Omega$ is a bounded domain containing the origin, $0\le \beta<\frac{N-ps}{2} $, $1

Solvable subclasses of a class of nonlinear second-order difference equations

Abstract: Abstract We present some solvable subclasses of the class of nonlinear second-order difference equations of the form Axn+1 + Bxn+1xn + Cxnxn-1 + Gxn+1xn-1 + Dxn + Exn-1 + F = 0, n ∈ ℕ0, where the parameters A, B, C, D, E, F, G and the initial values x-1,x0 are real numbers. This difference equation is a natural extension of the nonhomogeneous linear second-order difference equation with constant coefficients as well as of the bilinear difference equation with constant coefficients.

Periodic perturbations of Hamiltonian systems

Abstract: Abstract We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth

Abstract: Abstract We consider non-autonomous functionals of the form ℱ ⁢ ( u , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x {\\mathcal{F}(u,\\hskip-0.569055pt\\Omega)\\hskip-0.853583pt=\\hskip-0.853583pt\\int% _{\\Omega}f(x,\\hskip-0.569055ptDu(x))\\hskip-0.569055pt\\,dx} , where u : Ω → ℝ N {u\\colon\\kern-0.711319pt\\Omega\\hskip-0.569055pt\\to\\hskip-0.569055pt\\mathbb{R}^% {N}} , Ω ⊂ ℝ n {\\Omega\\subset\\mathbb{R}^{n}} . We assume that f ⁢ ( x , z ) {f(x,z)} grows at least as | z | p {|z|^{p}} and at most as | z | q {|z|^{q}} . Moreover, f ⁢ ( x , z ) {f(x,z)} is Hölder continuous with respect to x and convex with respect to z. In this setting, we give a sufficient condition on the density f ⁢ ( x , z ) {f(x,z)} that ensures the absence of a Lavrentiev gap.

The Brezis-Nirenberg problem for nonlocal systems

Abstract: By means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

Multi-bump solutions for a Kirchhoff-type problem

Abstract: Abstract In this paper, we study the existence of solutions for the Kirchhoff problem M(∫ ℝ 3 |∇u| 2 dx+∫ ℝ 3 (λa(x)+1)u 2 dx)(-Δu+(λa(x)+1)u)=f(u)$M\Biggl (\int _{\mathbb {R}^{3}}| abla u|^{2}\, dx + \int _{\mathbb {R}^{3}} (\lambda a(x)+1)u^{2}\, dx\Biggl ) (- \Delta u + (\lambda a(x)+1)u) = f(u)$ in ℝ3, u ∈ H1(ℝ3) Assuming that the nonnegative function a(x) has a potential well with int (a -1({0})) consisting of k disjoint components Ω1,Ω2,...,Ωk and the nonlinearity f(t) has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by using variational methods.

On sign-changing solutions for (p,q)-Laplace equations with two parameters

Abstract: We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous (p, q)-Laplace equations −Δpu − Δqu = α|u|p−2u + β|u|q−2u where p = q. By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the (α, β)-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension.

Maximum principles, Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations

Abstract: Abstract This paper is concerned with a general class of quasilinear anisotropic equations. We first derive some maximum principles for two appropriate P-functions, in the sense of Payne (see the book of Sperb [18]). These maximum principles are then employed to obtain a Liouville-type result and a Serrin–Weinberger-type symmetry result.

Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type

Abstract: Abstract We consider the existence of singular limit solutions for a nonlinear elliptic system of Liouville type with Dirichlet boundary conditions. We use the nonlinear domain decomposition method.

One-dimensional singular problems involving the p-Laplacian and nonlinearities indefinite in sign

Abstract: Let $\Omega$ be a bounded open interval, let $p>1$ and $\gamma>0$, and let $m:\Omega\rightarrow\mathbb{R}$ be a function that may change sign in $\Omega $. In this article we study the existence and nonexistence of positive solutions for one-dimensional singular problems of the form $-(\left\vert u^{\prime}\right\vert ^{p-2}u^{\prime})^{\prime}=m\left( x\right) u^{-\gamma}$ in $\Omega$, $u=0$ on $\partial\Omega$. As a consequence we also derive existence results for other related nonlinearities.

Existence results for nonlinear elliptic problems on fractal domains

Abstract: Some existence results for a parametric Dirichlet problem defined on the Sierpinski fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.

Unbounded solutions of third order three-point boundary value problems on a half-line

Abstract: Abstract We consider the following third order three-point boundary value problem on a half-line: x'''(t)+q(t)f(t,x(t),x'(t),x''(t)) = 0, t ∈ (0,+∞), x'(0) = A, x(η) = B, x''(+∞) = C, where η ∈ (0,+∞), but fixed, and f : [0,+∞) × ℝ3 → ℝ satisfies Nagumo's condition. We apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory, to establish existence theory for at least one unbounded solution, and at least three unbounded solutions. To demonstrate the usefulness of our results, we illustrate two examples.

On a class of fully nonlinear parabolic equations

Abstract: The first author was partially supported by the Research Project 15-11-20019 of the Russian Science Foundation (50% of all results of this paper). The second author acknowledges the support of the Research Grant MTM2013-43671-P, MICINN, Spain, and the program “Science Without Borders”, CSF-CAPES-PVE-Process 88887.059583/2014-00, Brasil.

Lichnerowicz-type equations on complete manifolds

Abstract: Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analysis.

Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

Abstract: Abstract In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

On the best constants in Sobolev inequalities on the solid torus in the limit case p = 1

Abstract: In this paper,we analyze the problem of determiningthe best constants for theSobolev inequalities inthelimitingcasewherep = 1.Firstly,thespecialcaseofthesolidtorusisstudied,wheneveritisprovedthat thesolidtorusisanextremaldomainwithrespecttothesecondbestconstantandtotallyoptimalwithrespect to the best constants in the trace Sobolev inequality. Secondly, in the spirit of Andreu, Mazon and Rossi (3), a Neumann problem involving the 1-Laplace operator in the solid torus is solved. Finally, the existence of bothbestconstantsinthecaseofamanifoldwithboundaryisstudied,whentheyexist.Furtherexamplesare provided where they do not exist. The impact of symmetries which appear in the manifold is also discussed.

Existence and global asymptotic behavior of positive solutions for combinedsecond-order differential equations on the half-line

Abstract: Abstract We are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem 1 A ⁢ ( A ⁢ u ′ ) ′ + a 1 ⁢ ( t ) ⁢ u σ 1 + a 2 ⁢ ( t ) ⁢ u σ 2 = 0 , t ∈ ( 0 , ∞ ) , $\\frac{1}{A}(Au^{\\prime})^{\\prime}+a_{1}(t)u^{\\sigma_{1}}+a_{2}(t)u^{\\sigma_{2}% }=0,\\quad t\\in(0,\\infty),$ subject to the boundary conditions lim t → 0 + ⁡ u ⁢ ( t ) = 0 ${\\lim_{t\\rightarrow 0^{+}}u(t)=0}$ , lim t → ∞ ⁡ u ⁢ ( t ) / ρ ⁢ ( t ) = 0 ${\\lim_{t\\rightarrow\\infty}{u(t)}/{\\rho(t)}=0}$ , where σ 1 , σ 2 < 1 ${\\sigma_{1},\\sigma_{2}<1}$ and A is a continuous function on [ 0 , ∞ ) ${[0,\\infty)}$ which is positive and differentiable on ( 0 , ∞ ) ${(0,\\infty)}$ such that ∫ 0 1 1 / A ⁢ ( t ) ⁢ 𝑑 t < ∞ ${\\int_{0}^{1}{1}/{A(t)}\\,dt<\\infty}$ and ∫ 0 ∞ 1 / A ⁢ ( t ) ⁢ 𝑑 t = ∞ ${\\int_{0}^{\\infty}{1}/{A(t)}\\,dt=\\infty}$ . Here, ρ ⁢ ( t ) = ∫ 0 t 1 / A ⁢ ( s ) ⁢ 𝑑 s ${\\rho(t)=\\int_{0}^{t}{1}/{A(s)}\\,ds}$ for t > 0 ${t>0}$ and a 1 , a 2 ${a_{1},a_{2}}$ are nonnegative continuous functions on ( 0 , ∞ ) ${(0,\\infty)}$ that may be singular at t = 0 ${t=0}$ and satisfying some appropriate assumptions related to the Karamata regular variation theory. Our approach is based on the sub-supersolution method.

An elliptic equation with an indefinite sublinear boundary condition

Abstract: Abstract We investigate the problem { - Δ ⁢ u = | u | p - 2 ⁢ u in Ω , ∂ ⁡ u ∂ ⁡ 𝐧 = λ ⁢ b ⁢ ( x ) ⁢ | u | q - 2 ⁢ u on ∂ ⁡ Ω , \\left\\{\\begin{aligned} \\displaystyle-\\Delta u&\\displaystyle=\\lvert u\\rvert^{p-% 2}u&&\\displaystyle\\phantom{}\\text{in ${\\Omega}$},\\\\ \\displaystyle\\frac{\\partial u}{\\partial\\mathbf{n}}&\\displaystyle=\\lambda b(x)% \\lvert u\\rvert^{q-2}u&&\\displaystyle\\phantom{}\\text{on ${\\partial\\Omega}$},% \\end{aligned}\\right. where Ω is a bounded and smooth domain of ℝ N {\\mathbb{R}^{N}} ( N ≥ 2 {N\\geq 2} ), 1 < q < 2 < p {1 0 {\\lambda>0} , and b ∈ C 1 + α ⁢ ( ∂ ⁡ Ω ) {b\\in C^{1+\\alpha}(\\partial\\Omega)} for some α ∈ ( 0 , 1 ) {\\alpha\\in(0,1)} . We show that ∫ ∂ ⁡ Ω b < 0 {\\int_{\\partial\\Omega}b<0} is a necessary and sufficient condition for the existence of nontrivial non-negative solutions of this problem. Under the additional condition b + ≢ 0 {b^{+}\ ot\\equiv 0} we show that for λ > 0 {\\lambda>0} sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in C ⁢ ( Ω ¯ ) {C(\\overline{\\Omega})} as λ → 0 {\\lambda\\to 0} . When p < 2 * {p<2^{*}} we also provide the asymptotic profiles of these solutions.

Radiative effects for some bidimensional thermoelectric problems

Abstract: Abstract There are two main objectives in this paper. One is to find sufficient conditions to ensure the existence of weak solutions for some bidimensional thermoelectric problems. At the steady-state, these problems consist of a coupled system of elliptic equations of the divergence form, commonly accomplished with nonlinear radiation-type conditions on at least a nonempty part of the boundary of a C 1 ${C^{1}}$ domain. The model under study takes the thermoelectric Peltier and Seebeck effects into account, which describe the Joule–Thomson effect. The proof method requires a fixed point argument. To this end, well-determined estimates are our main concern. The second objective of the paper is the derivation of explicit W 1 , p ${W^{1,p}}$ -estimates ( p > 2 ) ${(p>2)}$ for solutions of nonlinear radiation-type problems in the general n-dimensional space situation, where the leading coefficient is assumed to be a discontinuous function on the space variable. In particular, the behavior of the leading coefficient is conveniently explicit on the estimate of any solution.

Comparison and maximum principles for a class of flux-limited diffusions with external force fields

Abstract: In this paper, we are interested in a general equation that has finite speed of propagation compatible with Einstein's theory of special relativity. This equation without external force fields has been derived recently by means of optimal transportation theory. We first provide an argument to incorporate the external force fields. Then we are concerned with comparison and maximum principles for this equation. We consider both stationary and evolutionary problems. We show that the former satisfies a comparison principle and a strong maximum principle. While the latter fulfils weaker ones. The key technique is a transformation that matches well with the gradient flow structure of the equation.