Volume 18, Issue 1

We study the following coupled elliptic system with critical nonlinearities: { - △ ⁢ u + u = f ⁢ ( u ) + β ⁢ h ⁢ ( u ) ⁢ K ⁢ ( v ) , x ∈ ℝ N , - △ ⁢ v + v = g ⁢ ( v ) + β ⁢ H ⁢ ( u ) ⁢ k ⁢ ( v ) , x ∈ ℝ N , u , v ∈ H 1 ⁢ ( ℝ N ) , \left\{\begin{aligned} &\displaystyle-\triangle{u}+u=f(u)+\beta h(u)K(v),&&% \displaystyle x\in{\mathbb{R}}^{N},\\ &\displaystyle-\triangle{v}+v=g(v)+\beta H(u)k(v),&&\displaystyle x\in{\mathbb% {R}}^{N},\\ &\displaystyle u,v\in H^{1}({\mathbb{R}}^{N}),\end{aligned}\right. where β>0{\beta>0}; f , g are differentiable functions with critical growth; and H,K{H,K} are primitive functions of h and k , respectively. Under some assumptions on f , g , h and k , we obtain the existence of a positive ground state solution of this system for N≥2{N\geq 2}.

This paper is concerned with the following type of quasilinear elliptic equations in ℝN{\mathbb{R}^{N}} involving the p -Laplacian and critical growth: - Δ p ⁢ u + V ⁢ ( | x | ) ⁢ | u | p - 2 ⁢ u - Δ p ⁢ ( | u | 2 ) ⁢ u = λ ⁢ | u | q - 2 ⁢ u + | u | 2 ⁢ p * - 2 ⁢ u , -\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u, which arises as a model in mathematical physics, where 2<p<N{2<p<N}, p*=N⁢pN-p{p^{*}=\frac{Np}{N-p}}. For any given integer k≥0{k\geq 0}, by using change of variables and minimization arguments, we obtain, under some additional assumptions on p and q , a radial sign-changing nodal solution with k+1{k+1} nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

In this paper we analyze the Lane–Emden system - Δ ⁢ u = λ ⁢ f ⁢ ( x ) ( 1 - v ) 2 ⁢  in  ⁢ Ω , - Δ ⁢ v = μ ⁢ g ⁢ ( x ) ( 1 - u ) 2 ⁢  in  ⁢ Ω , 0 ≤ u , v < 1 ⁢  in  ⁢ Ω , u = v = 0 ⁢  on  ⁢ ∂ ⁡ Ω , -\Delta u=\frac{\lambda f(x)}{(1-v)^{2}}\text{ in }\Omega,\quad-\Delta v=\frac% {\mu g(x)}{(1-u)^{2}}\text{ in }\Omega,\quad 0\leq u,v<1\text{ in }\Omega,% \quad u=v=0\text{ on }\partial\Omega, where λ and μ are positive parameters and Ω is a smooth bounded domain of ℝN(N≥1){\mathbb{R}^{N}(N\geq 1)}. Here we prove the existence of a critical curve Γ which splits the positive quadrant of the (λ,μ){(\lambda,\mu)}-plane into two disjoint sets 𝒪1{\mathcal{O}_{1}} and 𝒪2{\mathcal{O}_{2}} such that the Lane–Emden system has a smooth minimal stable solution (uλ,vμ){(u_{\lambda},v_{\mu})} in 𝒪1{\mathcal{O}_{1}}, while for (λ,μ)∈𝒪2{(\lambda,\mu)\in\mathcal{O}_{2}} there are no solutions of any kind. We also establish upper and lower estimates for the critical curve Γ and regularity results on this curve if N≤7{N\leq 7}. Our proof is based on a delicate combination involving a maximum principle and Lp{L^{p}} estimates for semi-stable solutions of the Lane–Emden system.

We study a Klein–Gordon–Maxwell system in a bounded spatial domain under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many static solutions.

We prove structure results for the radial solutions of the semilinear problem Δ ⁢ u + λ ⁢ ( | x | ) | x | 2 ⁢ u + f ⁢ ( u ⁢ ( x ) , | x | ) = 0 , \Delta u+\frac{\lambda(|x|)}{|x|^{2}}u+f(u(x),|x|)=0, where λ is a function and f is superlinear in the u -variable. As particular cases, we are able to deal with Matukuma potentials and with nonlinearities f having different polynomial behaviors at zero and at infinity. We give the complete picture for the subcritical, critical and supercritical cases. The technique relies on the Fowler transformation, allowing to deal with a dynamical system in ℝ3{{\mathbb{R}}^{3}}, for which elementary invariant manifold theory allows to draw the conclusions involving regular/singular and fast/slow-decay solutions.

In this article, we study the existence of solutions for the Dirac system { D ⁢ u = ∂ ⁡ H ∂ ⁡ v ⁢ ( x , u , v )   on  ⁢ M , D ⁢ v = ∂ ⁡ H ∂ ⁡ u ⁢ ( x , u , v )   on  ⁢ M , \left\{\begin{aligned} \displaystyle Du&\displaystyle=\frac{\partial H}{% \partial v}(x,u,v)\quad\text{on }M,\\ \displaystyle Dv&\displaystyle=\frac{\partial H}{\partial u}(x,u,v)\quad\text{% on }M,\end{aligned}\right. where M is an m -dimensional compact Riemannian spin manifold, u,v∈C∞⁢(M,Σ⁢M){u,v\in C^{\infty}(M,\Sigma M)} are spinors, D is the Dirac operator on M , and the fiber preserving map H:Σ⁢M⊕Σ⁢M→ℝ{H:\Sigma M\oplus\Sigma M\rightarrow\mathbb{R}} is a real-valued superquadratic function of class C1{C^{1}} with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.

We consider Dirichlet elliptic equations driven by the sum of a p -Laplacian (2<p){(2<p)} and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both ±∞{\pm\infty} and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.

In this paper, we study the quasilinear Schrödinger equation -Δ⁢u+V⁢(x)⁢u-γ2⁢(Δ⁢u2)⁢u=|u|p-2⁢u{-\Delta u+V(x)u-\frac{\gamma}{2}(\Delta u^{2})u=|u|^{p-2}u}, x∈ℝN{x\in\mathbb{R}^{N}}, where V⁢(x):ℝN→ℝ{V(x):\mathbb{R}^{N}\to\mathbb{R}} is a given potential, γ>0{\gamma>0}, and either p∈(2,2*){p\in(2,2^{*})}, 2*=2⁢NN-2{2^{*}=\frac{2N}{N-2}} for N≥4{N\geq 4} or p∈(2,4){p\in(2,4)} for N=3{N=3}. If γ∈(0,γ0){\gamma\in(0,\gamma_{0})} for some γ0>0{\gamma_{0}>0}, we establish the existence of a positive solution uγ{u_{\gamma}} satisfying maxx∈ℝN⁡|γμ⁢uγ⁢(x)|→0{\max_{x\in\mathbb{R}^{N}}|\gamma^{\mu}u_{\gamma}(x)|\to 0} as γ→0+{\gamma\to 0^{+}} for any μ>12{\mu>\frac{1}{2}}. Particularly, if V⁢(x)=λ>0{V(x)=\lambda>0}, we prove the existence of a positive classical radial solution uγ{u_{\gamma}} and up to a subsequence, uγ→u0{u_{\gamma}\to u_{0}} in H2⁢(ℝN)∩C2⁢(ℝN){H^{2}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})} as γ→0+{\gamma\to 0^{+}}, where u0{u_{0}} is the ground state of the problem -Δ⁢u+λ⁢u=|u|p-2⁢u{-\Delta u+\lambda u=|u|^{p-2}u}, x∈ℝN{x\in\mathbb{R}^{N}}.

We study the solvability of a general class of cross diffusion systems and establish the local and global existence of their strong solutions under the weakest assumption that they are VMO. This work simplifies the setting in our previous work [15] and provides new extensions which are more verifiable in applications.

In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation x′′+f⁢(x)⁢x′+g⁢(t,x)=s{x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s}, where s is a real parameter, f and g are continuous functions and g is T -periodic in the variable t . The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions on g . Analogous results are also obtained for Neumann boundary conditions.

In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.