Positive solutions for a p−q‐Laplacian system with critical nonlinearities

In this paper, our main purpose is to establish the existence results of positive solutions for a p ?q ‐Laplacian system involving concave‐convex nonlinearities: where Ω is a bounded domain in R ^N , λ ,θ >0 and 1<r <q <p <N . We assume 1<α ,β and is the critical Sobolev exponent and △_s ·=div(|?·|^{s ?2}?·) is the s‐Laplacian operator. The main results are obtained by variational methods.     

Existence of positive solutions for nonlinear singular boundary value problem with p‐Laplacian

In this paper, we study the existence of positive solutions for the following Sturm–Liouville‐like four‐point singular boundary value problem (BVP) with p‐Laplacian where ?_p(s)=|s|^p?2 s, p>1, f is a lower semi‐continuous function. Using the fixed‐point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm–Liouville‐like singular BVP with p‐Laplacian are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.     

Existence of periodic solutions for the Newtonian equation of motion with p‐Laplacian operator

In this paper, we study the existence of periodic solutions for the Newtonian equation of motion with p ‐Laplacian operator by asymptotic behavior of potential function, establish some new sufficient criteria of existence of periodic solutions for the differential system under the frame of Fuc?ik spectrum, generalize and improve some known works, and give an example to illustrate the application of the theorems. Copyright © 2017 John Wiley & Sons, Ltd.     

Standing waves for a class of fractional p‐Laplacian equations with a general critical nonlinearity

In this paper, we concern with the following fractional p‐Laplacian equation with critical Sobolev exponent $$\left\{\begin{array}{l}{\epsilon }^{ps}{\left\{?\mathrm{\Delta }\right.}_p^{s}u+V(x){\left|u\right|}^{p?2}u=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u\text{in}{?}^N,\\ u\in W^{s,p}\left\{{?}^N\right.,u>0,\end{array}\right.$$ where ε > 0 is a small parameter,  λ > 0 , N is a positive integer, and N > ps with s ∈ (0, 1) fixed, $1<q\le p,p_s^{?}:=Np/\left(N?ps\right)$. Since the nonlinearity $h(x,u):=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u$ does not satisfy the following Ambrosetti‐Rabinowitz condition: $$0<\mu H(x,u):=\mu {\int }_0^{u}h(x,t)dt\le h(x,u)u,x\in {?}^N,0\ne u\in ?,$$ with μ > p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p‐Laplacian type.     

Removable isolated singularities of solutions to the weighted p‐Laplacian with singular convection

This paper is concerned with the removability of isolated singularities for the weighted p‐Laplacian with singular convection. Critical singular exponents included in diffusion and convection are exactly determined. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple solutions for a noncooperative Kirchhoff‐type system involving the fractional p‐Laplacian and critical exponents

In this paper, we use the Limit Index Theory due to Li 19 and the fractional version of concentration compactness principle to study the multiplicity of solutions for a class of noncooperative fractional p‐Laplacian elliptic system with homogeneous Dirichlet boundary conditions involving the critical exponents.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

Decay estimates for the wave equation of p‐Laplacian type with dissipation of m‐Laplacian type

In this paper, we study decay properties of solutions to the wave equation of p‐Laplacian type with a weak dissipation of m‐Laplacian type. Copyright © 2006 John Wiley & Sons, Ltd.     

Positive radial solutions of p‐Laplacian equation with sign changing nonlinear sources

This paper is concerned with the p‐Laplacian equation with singular sources which is allowed to change sign in a ball. The singularity may occur at the zero points of solutions and on the boundary. Using the upper and lower solutions method, we establish the existence of positive radial solutions for the problem considered. Copyright © 2006 John Wiley & Sons, Ltd.     

The 8π‐problem for radially symmetric solutions of a chemotaxis model in the plane

The existence, uniqueness and large time behaviour of radially symmetric solutions to a chemotaxis system in the plane ?² are studied either for the critical value of the mass equal to 8π or in the subcritical case. Copyright © 2006 John Wiley & Sons, Ltd.     

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results.     

New results on the existence of periodic solutions for a higher‐order Li énard type p‐Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher‐order Li énard type p‐Laplacian differential equation as follows Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Bifurcation and multiplicity results for critical fractional p‐Laplacian problems

We prove a bifurcation and multiplicity result for a critical fractional p‐Laplacian problem that is the analog of the Brézis‐Nirenberg problem for the nonlocal quasilinear case. This extends a result in the literature for the semilinear case to all , in particular, it gives a new existence result. When , the nonlinear operator , has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo‐index related to the ‐cohomological index that is applicable here.     

Positive solutions for multi‐point boundary value problems of fractional differential equations with p‐Laplacian

This paper investigates the existence of solutions for multi‐point boundary value problems of higher‐order nonlinear Caputo fractional differential equations with p‐Laplacian. Using the five functionals fixed‐point theorem, the existence of multiple positive solutions is proved. An example is also given to illustrate the effectiveness of ourmain result. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

In this paper, we prove the existence of ground state sign‐changing solutions for the following class of elliptic equation: where , and K(x) are positive continuous functions. Firstly, we obtain one ground state sign‐changing solution u_b by using some new analytical skills and non‐Nehari manifold method. Furthermore, the energy of u_b is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of u_b as the parameter b↘0. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence and multiplicity of solutions for critical elliptic equations with multi-polar potentials in symmetric domains

In this paper, we consider the elliptic equations with critical Sobolev exponents and multi-polar potentials in bounded symmetric domains and prove the existence and multiplicity of symmetric positive solutions by using the Ekeland variational principle and the Lusternik–Schnirelmann category theory.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2000 John Wiley & Sons, Ltd.