p-Laplacian Systems Research Articles

Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent

In this paper, we study the semilinear p-Laplacian system with critical growth terms in bounded domains. By using the Nehari manifold and variational methods, we prove that the system has at least two positive solutions when the pair of the parameters ( λ , μ ) belongs to a certain subset of R 2 .

A global multiplicity result for two-point boundary value problems of p-Laplacian systems

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for two-point boundary value problems of p-Laplacian systems which have a singular indefinite weight and real multiparameters. For proofs, we mainly make use of the upper and lower solution method and the fixed point index theorem. To obtain a global multiplicity result, we construct a weighted space to benefit richer topology of the solution space than C0-space.

PERIODIC SOLUTIONS FOR A KIND OF p-LAPLACIAN HAMILTONIAN SYSTEMS

In this paper, the existence of periodic solutions is obtained for a kind of p-Laplacian systems by the minimax methods in critical point theory. Moreover, the existence of infinite periodic solutions is also obtained.

Solutions and multiple solutions for periodic p-Laplacian systems with a non-smooth potential

In this article, we study non-linear periodic systems driven by the ordinary vector p-Laplacian differential operator and with a non-smooth potential. Using variational methods based on the non-smooth critical point theory, we prove an existence and a multiplicity theorems. The conditions on the non-smooth potential do not imply that the corresponding Euler functional of the problem is coercive.

Periodic and subharmonic solutions of discrete p-Laplacian systems

Some solvability conditions of periodic and subharmonic solutions are obtained for a class of discrete p-Laplacian systems by the minimax methods in critical point theory.

Existence and multiplicity of periodic solutions for the ordinary p-Laplacian systems

Some existence and multiplicity results are obtained for periodic solutions of the ordinary p-Laplacian systems: $$\left\{\begin{array}{@{}l@{\quad{}}l}(|u'(t)|^{p-2}u'(t))'=\nabla F(t,u(t)),&\mbox{a.e. }t\in[0,T],\\[4pt]u(0)-u(T)=u'(0)-u'(T)=0\end{array}\right.$$ by using the Saddle Point Theorem, the least action principle and the Three-critical-point Theorem.

Self-improving property of nonlinear higher order parabolic systems near the boundary

We establish global regularity results for a wide class of non-linear higher order parabolic systems. The model problem we have in mind is the parabolic p-Laplacian system of order 2m, m ≥ 1, $$\partial_t u + (-1)^{m}\, {\rm div}^m \left(|D^mu|^{p-2}D^{m}u\right) = 0$$ with prescribed boundary and initial values. We prove that if the boundary values are sufficiently regular, then Dmu is globally integrable to a better power than the natural p. The method also produces a global estimate.

Periodic solutions of non-autonomous ordinary p-Laplacian systems

The existence of periodic solutions are obtained for non-autonomous ordinary p-Laplacian systems by the least action principle in critical point theory.

Classes of infinite semipositone systems

We analyse the positive solutions to classes of p-Laplacian systems with Dirichlet boundary conditions, in particular, when the reaction terms tend to −∞ at the origin and satisfy a combined sublinear condition at ∞. We use the method of sub- and supersolutions to establish our results.

Multiple non-trivial solutions of the Neumann problem for p-Laplacian systems

We obtain multiple non-trivial solutions of the Neumann problem for p-Laplacian systems using Morse theory.

Periodic solutions of p-Laplacian systems with a nonlinear convection term

Periodic solutions of p-Laplacian systems with a nonlinear convection term

Existence of infinitely many periodic solutions for ordinary p-Laplacian systems

In this paper, we study the existence and multiplicity of non-trivial periodic solutions of ordinary p-Laplacian systems by using the minimax technique in critical point theory. We also give an example to illustrate that the obtained results are new even in the case p = 2 .

Boundary value problem for degenerate and singular p-laplacian systems

In this article, using the contraction mapping principle and the shooting method, the authors obtain the existence and uniqueness of the local solution and the global solution to a class of quasilinear elliptic systems with p-Laplacian as its principal. They also obtain the continuous dependence of the solutions on the boundary data.

Resonance problems for the p-Laplacian systems

Two existence theorems of the solutions are obtained for the p-Laplacian systems at resonance under a Landesman–Lazer-type condition by critical point theory.

Positive radial solutions for p-Laplacian systems

The paper deals with the existence of positive radial solutions for the p-Laplacian system div(|∇ ui|p-2∇ ui) + fi(u1, ..., un) = 0, |x| 1, x \in {{\mathbb{R}}}^N\) . Here fi, i = 1,...,n, are continuous and nonnegative functions. Let \({\bf u}= (u_1, \ldots, u_n), \| {\bf u}\| = \sum^n_{i=1}|u_i|, f^i_0 = \lim_{\| u \|\rightarrow 0} \frac{f^i(u)}{\| {\bf u}\| ^{p-1}}, f^i_\infty = \lim_{\| {\bf u}\| \rightarrow \infty} \frac{f^i(u)}{{\|\bf u}\| ^{p-1}}, i = 1, \ldots , n, {\bf f} = (f^1, \ldots , f^n), {\bf f}_0 = \sum^n_{i=1} f^i_0\) and f∞ = ∑ni=1fi∞. We prove that f0 = ∞ and f∞ = 0 (sublinear), guarantee the existence of positive radial solutions for the problem. Our methods employ fixed point theorems in a cone.

Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system

This paper deals with p-Laplacian systems u t - div ( | ∇ u | p - 2 ∇ u ) = a ∫ Ω u α 1 ( x , t ) v β 1 ( x , t ) d x , x ∈ Ω , t > 0 , v t - div ( | ∇ v | q - 2 ∇ v ) = b ∫ Ω u α 2 ( x , t ) v β 2 ( x , t ) d x , x ∈ Ω , t > 0 with null Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ R N , where p , q > 1 , α i , β i ⩾ 0 , i = 1 , 2 , and a , b > 0 are positive constants. We first get the non-existence result for a related elliptic systems of non-increasing positive solutions. Secondly by using this non-existence result, blow-up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω = B R = { x ∈ R N : | x | < R } ( R > 0 ) . Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exists globally or blow-up in finite time.

Gradient estimates for a class of parabolic systems

We establish local Calderón-Zygmund-type estimates for a class of parabolic problems whose model is the nonhomogeneous, degenerate/singular parabolic p-Laplacian system ut-div(|Du|p-2Du)=div(|F|p-2F), proving that F∈Llocq⇒Du∈Lqloc, ∀ q≥p. We also treat systems with discontinuous coefficients of vanishing mean oscillation (VMO) type

Positive solutions for a class of p-Laplacian systems with multiple parameters

Consider the system − Δ p u = λ 1 f ( v ) + μ 1 h ( u ) in Ω , − Δ q v = λ 2 g ( u ) + μ 2 γ ( v ) in Ω , u = 0 = v on ∂ Ω , where Δ s z = div ( | ∇ z | s − 2 ∇ z ) , s > 1 , λ 1 , λ 2 , μ 1 and μ 2 are nonnegative parameters, and Ω is a bounded domain in R N with smooth boundary ∂ Ω. We prove the existence of a large positive solution for λ 1 + μ 1 and λ 2 + μ 2 large when lim x → ∞ f ( M [ g ( x ) ] 1 / q − 1 ) x p − 1 = 0 for every M > 0 , lim x → ∞ h ( x ) x p − 1 = 0 and lim x → ∞ γ ( x ) x q − 1 = 0 . In particular, we do not assume any sign conditions on f ( 0 ) , g ( 0 ) , h ( 0 ) or γ ( 0 ) . We also discuss a multiplicity results when f ( 0 ) = g ( 0 ) = h ( 0 ) = γ ( 0 ) = 0 .

Structure of Coexistence States for a Class of Quasilinear Elliptic Systems

The structure of positive solutions of the p-Laplacian systems is discussed via bifurcation theory and monotone techniques.

Some existence results on periodic solutions of ordinary p-Laplacian systems

Some existence theorems are obtained for periodic solutions of ordinary p-Laplacian systems by the minimax methods in critical point theory.