Positive solutions for critical quasilinear elliptic equations with mixed dirichlet-neumann boundary conditions

The existence and multiplicity of positive solutions are studied for a class of quasilinear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.     

Positive solutions for logistic type quasilinear elliptic equations on R

In this paper, we consider positive solutions of the logistic type p-Laplacian equation −Δ_pu=a(x)|u|^p−2u−b(x)|u|^q−1u, xR^N (N2). We show that under rather general conditions on a(x) and b(x) for large |x|, the behavior of the positive solutions for large |x| can be determined. This is then used to show that there is a unique positive solution. Our results improve the corresponding ones in J. London Math. Soc. (2) 64 (2001) 107–124 and J. Anal. Math., in press.     

Generalized periodic solutions of quasilinear equations

We study a boundary-value periodic problem for the quasilinear equationu _ff ?u _xx =F[u,u _f u _x ],u(0,t) =u (π,t),u (x, t + π/q) =u(x, t), 0 ≤x ≤π,t ∈ ?,q ∈ ?. We establish conditions under which the theorem on the uniqueness of a smooth solution is true.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth

This paper is concerned with the positive solutions for generalized quasilinear Schrödinger equations in R^N${\mathbb{R}}^N$ with critical growth which have appeared from plasma physics, as well as high-power ultrashort laser in matter. By using a change of variables and variational argument, we obtain the existence of positive solutions for the given problem.     

Multibump solutions for quasilinear elliptic equations

The current paper is concerned with constructing multibump type solutions for a class of quasilinear Schrödinger type equations including the Modified Nonlinear Schrödinger Equations. Our results extend the existence results on multibump type solutions in Coti Zelati and Rabinowitz (1992) [17] to the quasilinear case. Our work provides a theoretic framework for dealing with quasilinear problems, which lack both smoothness and compactness, by using more refined variational techniques such as gluing techniques, Morse theory, Lyapunov–Schmidt reduction, etc.     

On oscillatory solutions of quasilinear differential equations

Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero-counting function.     

Stability of solutions of quasilinear parabolic equations

We bound the difference between solutions u and v of ut=aΔu+divxf+h and vt=bΔv+divxg+k with initial data φ and ψ, respectively, by

     

Global solutions of abstract quasilinear parabolic equations

Local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied. Applications to quasilinear parabolic partial differential equations are also given.     

Periodic solutions of second-order quasilinear hyperbolic equations

We study a periodic boundary-value problem for a quasilinear equation with the d'Alembert operator on the left-hand side and a nonlinear operator on the right-hand side and establish conditions under which the solution of the indicated problem is unique.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1370–1375, October, 1995.     

Modulating pulse solutions for quasilinear wave equations

This paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave train; they are also referred to as ‘moving breathers’ since they are time periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with two stable, two unstable and infinitely many neutral directions. Using a partial normal form and a generalisation of local invariant-manifold theory to the quasilinear setting we prove the existence of modulating pulses on arbitrarily large, but finite domains in space and time.