Positive Smooth Solutions Research Articles

Weak solutions to lubrication systems describing the evolution of bilayer thin films

The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

A pair of positive solutions for $(p,q)$-equations with combined nonlinearities

We consider a parametric nonlinear Dirichlet problem driven by the $(p,q)$-differential operator, with a reaction consisting of a ``concave'' term perturbed by a $(p-1)$-superlinear perturbation, which need not satisfy the Ambrosetti-Rabinowitz condition (problem with combined or competing nonlinearities). Using variational methods we show that for small values of the parameter the problem has at least two nontrivial positive smooth solutions.

P-Laplace equations with singular terms and p-superlinear perturbations

We consider a nonlinear parametric Neumann problem driven by the p-Laplacian with a singular term and a Caratheodory perturbation whose primitive is p- superlinear near +∞ but needs not satisfy the Ambrosetti-Rabinowitz condition. We show that when the parameter λ>0 is small, the problem has at least two nontrivial positive smooth solutions. Our method of proof uses an approximation of the original problem, which we solve using variational techniques and then pass to the limit.

Self-similar solutions of the Euler equations with spherical symmetry

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.

Existence and uniqueness of positive solutions for the Neumann p-Laplacian

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Caratheodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions.

A bifurcation theorem for the p-logistic equation

We consider a p-logistic equation with an equidiffusive reaction. Using variational methods and truncation techniques, we show that there is a critical parameter value λ ∗ > 0 such that for λ > λ ∗ the problem has a unique positive smooth solution, and for λ ∈ (0, λ ∗] the problem has no positive solution.

Multiple positive solutions for a class of [formula omitted]-superlinear elliptic problems

We study nonlinear elliptic Dirichlet problems driven by the p -Laplacian and with a nonlinearity which consists of a p -superlinear term perturbed by a p -linear one. Using critical point theory, coupled with suitable truncation techniques, we show that the problem has two positive smooth solutions. When the conditions are symmetric in the two semiaxes, we have four nontrivial smooth solutions of constant sign. Finally for the semilinear problem, using Morse theory, we produce a fifth nontrivial smooth solution.

Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space

In this Note, we give a uniform bound and a non-existence result for positive solutions to the Lichnerowicz equation in R n . In particular, we show that positive smooth solutions to: Δ u + f ( u ) = 0 , u > 0 , in R n where f ( u ) = u − p − 1 − u p − 1 , are uniformly bounded. To cite this article: L. Ma, X. Xu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Existence of a pair of positive solutions for a class of nonlinear eigenvalue problems

In this paper, we consider a nonlinear elliptic equation driven by the p-Laplacian and with a parameter λ > 0. Using a combination of variational and degree theoretic methods, we show that there exists λ* > 0 such that, if λ > λ*, then the problem has two positive smooth solutions. Our result extends earlier ones by Rabinowitz (semilinear equations) and Guo (nonlinear equations).

Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities

We consider a nonlinear Dirichlet problem driven by the $p$--Laplacian differential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is $p$--linear and resonant with respect to $\lambda_1>0$ (the principal eigenvalue of $(-\Delta_p,W^{1,p}_0(Z))$) at infinity and the other when the perturbation is $p$--superlinear at infinity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper--lower solutions and with suitable truncation techniques.

Pairs of positive solutions for [formula omitted]-Laplacian equations with sublinear and superlinear nonlinearities which do not satisfy the AR-condition

We consider a nonlinear Dirichlet problem driven by the p -Laplacian differential. The right-hand-side nonlinearity, exhibits a ( p − 1 ) -sublinear term of the form m ( z ) | x | r − 2 x , r < p (concave term), and a Carathéodory term f ( z , x ) which is ( p − 1 ) -superlinear near + ∞ . However, it does not satisfy the usual Ambrosetti–Rabinowitz condition (AR-condition). Instead we employ a more general condition. Using a variational approach based on the critical point theory and the Ekeland variational principle, we show the existence of two nontrivial positive smooth solutions and then the existence of two nontrivial negative smooth solutions.

The compactness theorem for inhomogeneous semilinear elliptic equations with supercritical exponents

In this paper, we consider the following inhomogeneous semilinear equation: Δ u + k ( x ) u α + f ( x ) = 0 in Ω ⊂ R n ( n ≥ 3 ) where α ≥ n + 2 n − 2 , k ( x ) , f ( x ) ∈ C 1 ( Ω ) ; a 1 < k ( x ) < b 1 , 0 < a 1 < b 1 and | ∇ log k ( x ) | ≤ C for x ∈ Ω ¯ ; f ( x ) > 0 , ‖ f ‖ L ∞ ( Ω ) ≤ a 2 and ‖ ∇ f ‖ L ∞ ( Ω ) ≤ b 2 for some constants a 2 , b 2 > 0 . Using the monotonicity inequality and ε -regularity result, we obtain the measure estimate of the blow up set for positive smooth solutions { u i } of the above equation with { ‖ u i ‖ H 1 ( Ω ) + ‖ u i ‖ L α + 1 ( Ω ) } bounded.

Positive solutions and multiple solutions at non-resonance, resonance and near resonance for hemivariational inequalities with 𝑝-Laplacian

In this paper we study eigenvalue problems for hemivariational inequalities driven by thepp-Laplacian differential operator. We prove the existence of positive smooth solutions for both non-resonant and resonant problems at the principal eigenvalue of the negativepp-Laplacian with homogeneous Dirichlet boundary condition. We also examine problems which are near resonance both from the left and from the right of the principal eigenvalue. For nearly resonant from the right problems we also prove a multiplicity result.

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L 1 lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.

Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities

AbstractIn this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Nonlinear Dynamics of Electrified Thin Liquid Films

We study a nonlinear nonlocal evolution equation describing the hydrodynamics of thin films in the presence of normal electric fields. The liquid film is assumed to be perfectly conducting and to completely wet the upper or lower surface of a horizontal flat plate. The flat plate is held at constant voltage, and a vertical electric field is generated by a second parallel electrode kept at a different constant voltage and placed at a large vertical distance from the bottom plate. The fluid is viscous, and gravity and surface tension act. The equation is derived using lubrication theory and contains an additional nonlinear nonlocal term representing the electric field. The electric field is linearly destabilizing and is particularly important in producing nontrivial dynamics in the case when the film rests on the upper side of the plate. We give rigorous results on the global boundedness of positive periodic smooth solutions, using an appropriate energy functional. We also implement a fully implicit numerical scheme and perform extensive numerical experiments. Through a combination of analysis and numerical experiments we present evidence for the global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena, which are also observed in hanging films when electric fields are absent.

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H10 versus C10 minimizers of a C1-functional.

On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds

We study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα was introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn,[g]). We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than (n−α)/2 for some α∈[2,n), the set of positive smooth solutions to the equation Pαu=u(n+α)/(n−α) is compact in the C∞ topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory.

Positive Solution of Singular Boundary Value Problems on a Half-Line

This paper investigates existence of positive solutions of singular sub-linear boundary value problems on a half-line. Necessary and sufficient conditions for the existence of positive continuous solutions or smooth solutions on [0, ∞) are given by constructing new lower and upper solutions.

Solvability of nonlinear variational–hemivariational inequalities

In this paper we study nonlinear elliptic differential equations driven by the p-Laplacian with unilateral constraints produced by the combined effects of a monotone term and of a nonmonotone term (variational–hemivariational inequality). Our approach is variational and uses the subdifferential theory of nonsmooth functions and the theory of accretive and monotone operators. Also using these ideas and a special choice of the monotone term, we prove the existence of a strictly positive smooth solution for a class of nonlinear equations with nonsmooth potential (hemivariational inequality).