Volume 19, Issue 3

This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem { - ( u ′ 1 + u ′ 2 ) ′ = λ ⁢ a ⁢ ( x ) ⁢ f ⁢ ( u )   in  ⁢ ( 0 , 1 ) , u ′ ⁢ ( 0 ) = 0 , u ′ ⁢ ( 1 ) = 0 , \begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{% \prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\ u^{\prime}(0)=0,\quad u^{\prime}(1)=0,\end{dcases} where λ∈ℝ{\lambda\in\mathbb{R}} is a parameter, a∈L∞⁢(0,1){a\in L^{\infty}(0,1)} changes sign, and f∈C1⁢(ℝ){f\in C^{1}(\mathbb{R})} is positive in (0,+∞){(0,+\infty)}. The attention is focused on the case f⁢(0)=0{f(0)=0} and f′⁢(0)=1{f^{\prime}(0)=1}, where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a , as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space: ∫ ℝ + n ∫ ∂ ⁡ ℝ + n | x | α | x - y | λ f ( x ) g ( y ) | y | β d y d x ≥ C n , α , β , p , q ′ ∥ f ∥ L q ′ ⁢ ( ℝ + n ) ∥ g ∥ L p ⁢ ( ∂ ⁡ ℝ + n ) \int_{\mathbb{R}^{n}_{+}}\int_{\partial\mathbb{R}^{n}_{+}}\lvert x|^{\alpha}|x% -y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\|% f\/\|_{L^{q^{\prime}}(\mathbb{R}^{n}_{+})}\|g\|_{L^{p}(\partial\mathbb{R}^{n}_% {+})} for any nonnegative functions f∈Lq′⁢(ℝ+n){f\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})}, g∈Lp⁢(∂⁡ℝ+n){g\in L^{p}(\partial\mathbb{R}^{n}_{+})}, and p,q′∈(0,1){p,q^{\prime}\in(0,1)}, β<1-np′{\beta<\frac{1-n}{p^{\prime}}} or α<-nq{\alpha<-\frac{n}{q}}, λ>0{\lambda>0} satisfying n - 1 n ⁢ 1 p + 1 q ′ - α + β + λ - 1 n = 2 . \frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n}% =2. Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space ℝ+n{\mathbb{R}^{n}_{+}}.

In this paper, we prove that on every Finsler manifold (M,F){(M,F)} with reversibility λ and flag curvature K satisfying (λλ+1)2<K≤1{(\frac{\lambda}{\lambda+1})^{2}<K\leq 1}, there exist [dim⁡M+12]{[\frac{\dim M+1}{2}]} closed geodesics. If the number of closed geodesics is finite, then there exist [dim⁡M2]{[\frac{\dim M}{2}]} non-hyperbolic closed geodesics. Moreover, there are three closed geodesics on (M,F){(M,F)} satisfying the above pinching condition when dim⁡M=3{\dim M=3}.

We show that the E -cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals on Hilbert spaces. When applied to the setting of the Arnold conjecture, this paves the way to a short proof on tori, where it was first shown by C. Conley and E. Zehnder in 1983.

This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an n -dimensional ball. By combining the variational methods and saddle point reduction technique, we obtain the existence of at least three periodic solutions for arbitrary space dimension n . The structure of the spectrum of the linearized problem plays an essential role in the proof, and the construction of a suitable working space is devised to overcome the restriction of space dimension.

An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite-dimensional Hilbert Spaces. The applications are given to Schrödinger equations. Here spectral properties inherited by the potential features are exploited in order to establish a linking structure, and hence hypotheses of monotonicity on the nonlinearities are discarded.

We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions are mildly localized as they lie in the essential spectrum of the corresponding linear operator. As a new auxiliary tool a limiting absorption principle for the curl-curl operator is proved.

Consider the system { - Δ ⁢ u i + μ i ⁢ u i = ν i ⁢ u i 2 * - 1 + β ⁢ ∑ j = 1 , j ≠ i k u j 2 * 2 ⁢ u i 2 * 2 - 1 + λ ⁢ ∑ j = 1 , j ≠ i k u j in ⁢ Ω , u i > 0 in ⁢ Ω , u i = 0 on ⁢ ∂ ⁡ Ω , i = 1 , 2 , … , k , \left\{\begin{aligned} \displaystyle-\Delta u_{i}+\mu_{i}u_{i}&\displaystyle=% \nu_{i}u_{i}^{2^{*}-1}+\beta\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}^{\frac{2^{*}}% {2}}u_{i}^{\frac{2^{*}}{2}-1}+\lambda\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}&&% \displaystyle\phantom{}\text{in}\ \Omega,\\ \displaystyle u_{i}&\displaystyle>0&&\displaystyle\phantom{}\text{in}\ \Omega,% \\ \displaystyle u_{i}&\displaystyle=0&&\displaystyle\phantom{}\text{on}\ % \partial\Omega,\quad i=1,2,\ldots,k,\end{aligned}\right. where k≥2{k\geq 2}, Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} (N≥3{N\geq 3}) is a bounded domain, 2*=2⁢NN-2{2^{*}=\frac{2N}{N-2}}, μi∈ℝ{\mu_{i}\in\mathbb{R}} and νi>0{\nu_{i}>0} are constants, and β,λ>0{\beta,\lambda>0} are parameters. By showing a unique result of the limit system, we prove existence and nonexistence results of ground states to this system by variational methods, which generalize the results in [7, 18]. Concentration behaviors of ground states for β,λ{\beta,\lambda} are also established.

In this paper, we consider the following class of differential inclusion problems in ℝN{\mathbb{R}^{N}} involving the p⁢(x){p(x)}-Laplacian: - Δ p ⁢ ( x ) ⁢ u + V ⁢ ( x ) ⁢ | u | p ⁢ ( x ) - 2 ⁢ u ∈ a ⁢ ( x ) ⁢ ∂ ⁡ F ⁢ ( x , u )   in ⁢ ℝ N . -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.