Abstract This article is a survey on boundedness results for stable solutions to semilinear elliptic problems.For these solutions, we present the currently known L ∞ ${L^{\infty}}$ estimates that hold for all nonlinearities.Such estimates are known to hold up to dimension 4.They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only beenestablished in the radial case.

Abstract We deal with very weak positive supersolutions to the Hénon–Lane–Emden system on neighborhoods of the origin. In our main theorem we prove a sharp nonexistence result.

Abstract We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equations - Δ ⁢ u + u = ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u ) in ⁢ ℝ 2 , -\Delta u+u=\bigl{(}I_{\alpha}*F(u)\bigr{)}F^{\prime}(u)\quad\text{in }\mathbb% {R}^{2}, where I α {I_{\alpha}} is the Riesz potential of order α on the plane ℝ 2 {\mathbb{R}^{2}} under general nontriviality, growth and subcriticality on the nonlinearity F ∈ C 1 ⁢ ( ℝ , ℝ ) {F\in C^{1}(\mathbb{R},\mathbb{R})} .

Abstract We establish the existence of nodal solutions to the supercritical problem - Δ ⁢ u = | u | p - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $-\Delta u=|u|^{p-2}u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ in a symmetric bounded smooth domain Ω of ℝ N ${\mathbb{R}^{N}}$ , N ≥ 3 ${N\geq 3}$ , for p > 2 N ∗ := 2 ⁢ N N - 2 ${p>2_{N}^{\ast}:=\frac{2N}{N-2}}$ , up to some range which depends on the symmetries, and we study their asymptotic behavior as p → 2 N ∗ ${p\rightarrow 2_{N}^{\ast}}$ . We exhibit solutions u p ${u_{p}}$ to this problem in symmetric domains with a shrinking hole, which concentrate at a single point as the hole shrinks and p approaches 2 N ∗ ${2_{N}^{\ast}}$ from above, and whose limit profile is a rescaling of a nonradial sign-changing solution to the limit problem - Δ ⁢ u = | u | 2 N ∗ - 2 ⁢ u , u ∈ D 1 , 2 ⁢ ( ℝ N ) . $-\Delta u=|u|^{2_{N}^{\ast}-2}u,\quad u\in D^{1,2}(\mathbb{R}^{N}).$

Abstract We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range ( e 0 , e 1 ) ${(e_{0},e_{1})}$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, e 0 = 0 ${e_{0}=0}$ is the minimal energy of the system).

Abstract In this paper, we study the following biharmonic equations: { Δ 2 ⁢ u - a 0 ⁢ Δ ⁢ u + ( λ ⁢ b ⁢ ( x ) + b 0 ) ⁢ u = f ⁢ ( u ) in ⁢ ℝ N , u ∈ H 2 ⁢ ( ℝ N ) , $\left\{\begin{aligned} &\displaystyle\Delta^{2}u-a_{0}\Delta u+(\lambda b(x)+b% _{0})u=f(u)&\hskip 10.0pt\text{in }\mathbb{R}^{N},\\ &\displaystyle u\in H^{2}(\mathbb{R}^{N}),\end{aligned}\right.$ where N ≥ 3 ${N\geq 3}$ , a 0 , b 0 ∈ ℝ ${a_{0},b_{0}\in\mathbb{R}}$ are two constants, λ > 0 ${\lambda>0}$ is a parameter, b ⁢ ( x ) ≥ 0 ${b(x)\geq 0}$ is a potential well and f ⁢ ( t ) ∈ C ⁢ ( ℝ ) ${f(t)\in C(\mathbb{R})}$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo–Nirenberg inequality, we make some observations on the operator Δ 2 - a 0 ⁢ Δ + λ ⁢ b ⁢ ( x ) + b 0 ${\Delta^{2}-a_{0}\Delta+\lambda b(x)+b_{0}}$ in H 2 ⁢ ( ℝ N ) ${H^{2}(\mathbb{R}^{N})}$ . Based on these observations, we give a new variational setting to the above problem for a 0 < 0 ${a_{0}<0}$ . With this new variational setting in hands, we establish some new existence results of the nontrivial solutions for all a 0 < 0 ${a_{0}<0}$ with λ sufficiently large by the variational method. The concentration behavior of the nontrivial solution for λ → + ∞ ${\lambda\to+\infty}$ is also obtained. It is worth pointing out that it seems to be the first time that the nontrivial solution of the above problem is obtained for all a 0 < 0 ${a_{0}<0}$ .

Abstract In this paper, we deal with the singular perturbed fractional elliptic problem ε ⁢ ( - Δ ) 1 / 2 ⁢ u + V ⁢ ( z ) ⁢ u = f ⁢ ( u ) $\varepsilon(-\Delta)^{1/2}{u}+V(z)u=f(u)$ in ℝ $\mathbb{R}$ , where ( - Δ ) 1 / 2 ⁢ u ${(-\Delta)^{1/2}u}$ is the square root of the Laplacian and f ⁢ ( s ) ${f(s)}$ has exponential critical growth. Under suitable conditions on f ⁢ ( s ) ${f(s)}$ , we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of V as ε goes to 0.

Abstract In this paper we consider a nonlinear elliptic problem driven by a nonhomogeneous differential operator with Robin boundary conditions. We produce conditions on the reaction term near 0 + ${0^{+}}$ and near + ∞ ${+\infty}$ , which imply the existence and uniqueness of a positive solution. In the particular case of equations driven by the p-Laplacian, we show that these conditions are also necessary, extending in this way the semilinear Dirichlet work of Brezis and Oswald [5].

Abstract This paper is focused on the local interior W 1 , ∞ ${W^{1,\infty}}$ -regularity for weak solutions of degenerate elliptic equations of the form div ⁡ [ 𝐚 ⁢ ( x , u , ∇ ⁡ u ) ] + b ⁢ ( x , u , ∇ ⁡ u ) = 0 ${\operatorname{div}[\mathbf{a}(x,u,\nabla u)]+b(x,u,\nabla u)=0}$ , which include those of p-Laplacian type. We derive an explicit estimate of the local L ∞ ${L^{\infty}}$ -norm for the solution’s gradient in terms of its local L p ${L^{p}}$ -norm. Specifically, we prove ∥ ∇ ⁡ u ∥ L ∞ ⁢ ( B R / 2 ⁢ ( x 0 ) ) p ≤ C | B R ⁢ ( x 0 ) | ⁢ ∫ B R ⁢ ( x 0 ) | ∇ ⁡ u ⁢ ( x ) | p ⁢ 𝑑 x . $\lVert\nabla u\rVert_{L^{\infty}(B_{R/2}(x_{0}))}^{p}\leq\frac{C}{\lvert B_{R}% (x_{0})\rvert}\int_{B_{R}(x_{0})}\lvert\nabla u(x)\rvert^{p}dx.$ This estimate paves the way for our work [9] in establishing W 1 , q ${W^{1,q}}$ -estimates (for q > p ${q>p}$ ) for weak solutions to a much larger class of quasilinear elliptic equations.

Abstract We prove the presence of chaotic dynamics for the classical two-body Kepler problem with a time-periodic gravitational coefficient oscillating between two fixed values. The set of chaotic solutions we detect is coded by the number of revolutions in each period. The chaotic dynamics is obtained for large period T as well as for small angular momentum μ. In particular, we provide an explicit lower bound on T and explicit upper bound on μ which guarantee the existence of complex dynamics. We get our results by applying a simple and well-known topological method, the stretching along the path technique. Our results are robust with respect to small perturbations of the gravitational coefficient and to the addition of a small friction term.