Volume 20, Issue 1

In this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies - Δ ⁢ u = f ⁢ ( u )   in  ⁢ B 1 , -\Delta u=f(u)\quad\text{in }B_{1}, where u∈C2⁢(B1){u\in C^{2}(B_{1})}, B1{B_{1}} is the unit ball of ℝ2{\mathbb{R}^{2}} and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.

This paper deals with nonlinear elliptic problems where the diffusion coefficient is a degenerate non-local term. We show that this degeneration implies the growth of the complexity of the structure of the set of positive solutions of the equation. Specifically, when the reaction term is of logistic type, the continuum of positive solutions breaks into two disjoint pieces. Our approach uses mainly fixed point arguments.

This article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

In this paper, we study the following nonlinear Schrödinger–Newton type system: { - ϵ 2 ⁢ Δ ⁢ u + u - Φ ⁢ ( x ) ⁢ u = Q ⁢ ( x ) ⁢ | u | ⁢ u , x ∈ ℝ 3 , - ϵ 2 ⁢ Δ ⁢ Φ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right. where ϵ>0{\epsilon>0} and Q⁢(x){Q(x)} is a positive bounded continuous potential on ℝ3{\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k , the system has a positive solution with k -peaks concentrating near a strict local minimum point x0{x_{0}} of Q⁢(x){Q(x)} in ℝ3{\mathbb{R}^{3}}, provided that ϵ>0{\epsilon>0} is sufficiently small.

This paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models -△⁢u±λ⁢|∇⁡u|2uβ=b⁢(x)⁢u-α{-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, u>0{u>0}, x∈Ω{x\in\Omega}, u|∂⁡Ω=0{u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in ℝN{\mathbb{R}^{N}}, λ>0{\lambda>0}, β>0{\beta>0}, α>-1{\alpha>-1}, and b∈Clocν⁢(Ω){b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some ν∈(0,1){\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on ∂⁡Ω{\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

The main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups G (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case G is an Iwasawa group, it is shown here that the L2{L^{2}}-CKN inequalities are sharp for all parameter values except one exceptional case. To show this, generalized Kelvin transforms Kσ{K_{\sigma}} are introduced and shown to be isometries for certain weighted Sobolev spaces. An interesting transformation formula for the sub-Laplacian with respect to Kσ{K_{\sigma}} is also derived. Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.

In this paper, we establish a new singular Trudinger–Moser type inequality for radial Sobolev spaces with logarithmic weights. The existence of nontrivial solutions is proved for an elliptic equation defined in ℝn{\mathbb{R}^{n}}, relying on variational methods and involving a nonlinearity with doubly exponential growth at infinity.

We consider the following Choquard equation: { - Δ ⁢ u = ( ∫ Ω | u ⁢ ( y ) | 2 μ * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | 2 μ * - 2 ⁢ u in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\Bigg{(}\int_{% \Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,dy\Bigg{)}|u|^{2^{*}_{\mu}-2}% u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth bounded domain in ℝN{\mathbb{R}^{N}} (N≥3{N\geq 3}), 2μ*=2⁢N-μN-2{2^{*}_{\mu}=\frac{2N-\mu}{N-2}}. This paper is concerned with the existence of a positive high-energy solution of the above problem in an annular-type domain when the inner hole is sufficiently small.

In this paper, we consider fractional Choquard equations with confining potentials. First, we show that they admit a positive ground state and infinitely many bound states. Then we prove the existence of two signed solutions when a superlinear and subcritical perturbation is added; in this case, the main feature is that such a perturbation does not satisfy the usual Ambrosetti–Rabinowitz condition.

We consider the following fractional Schrödinger equation involving critical exponent: { ( - Δ ) s ⁢ u + V ⁢ ( y ) ⁢ u = u 2 s * - 1 in  ⁢ ℝ N , u > 0 , y ∈ ℝ N , \left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u+V(y)u=u^{2^{*}_{s}-1}&&% \displaystyle\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,&&\displaystyle y\in\mathbb{R}^{N},\end{aligned}\right. where N≥3{N\geq 3} and 2s*=2⁢NN-2⁢s{2^{*}_{s}=\frac{2N}{N-2s}} is the critical Sobolev exponent. Under some suitable assumptions of the potential function V⁢(y){V(y)}, by using a finite-dimensional reduction method, combined with various local Pohazaev identities, we prove the existence of infinitely many solutions. Due to the nonlocality of the fractional Laplacian operator, we need to study the corresponding harmonic extension problem.

We consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator, in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such an operator generally enjoys the qualitative properties of the Laplace operator, as for instance maximum and comparison principles, ABP and Harnack inequalities, Liouville theorems for subsolutions or supersolutions. Existence and uniqueness for the Dirichlet problem are also proved as well as local and global Hölder estimates for viscosity solutions. All results are discussed for a more general class of weighted partial trace operators.