Volume 21, Issue 4

The main purpose of this paper is to establish the existence of ground-state solutions to a class of Schrödinger equations with critical exponential growth involving the nonnegative, possibly degenerate, potential V: - div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) . -\operatorname{div}(\lvert\nabla u\rvert^{n-2}\nabla u)+V(x)\lvert u\rvert^{n-% 2}u=f(u). To this end, we first need to prove a sharp Trudinger–Moser inequality in ℝn{\mathbb{R}^{n}} under the constraint ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 . \int_{\mathbb{R}^{n}}(\lvert\nabla u\rvert^{n}+V(x)\lvert u\rvert^{n})\,dx\leq 1. This is proved without using the technique of blow-up analysis or symmetrization argument. As far as what has been studied in the literature, having a positive lower bound has become a standard assumption on the potential V⁢(x){V(x)} in dealing with the existence of solutions to the above Schrödinger equation. Since V⁢(x){V(x)} is allowed to vanish on an open set in ℝn{\mathbb{R}^{n}}, the loss of a positive lower bound of the potential V⁢(x){V(x)} makes this problem become fairly nontrivial. Our method to prove the Trudinger–Moser inequality in ℝ2{\mathbb{R}^{2}} (see [L. Chen, G. Lu and M. Zhu, A critical Trudinger–Moser inequality involving a degenerate potential and nonlinear Schrödinger equations, Sci. China Math. 64 2021, 7, 1391–1410]) does not apply to this higher-dimensional case ℝn{\mathbb{R}^{n}} for n≥3{n\geq 3} here. To obtain the existence of a ground state solution, we use a non-symmetric argument to exclude the possibilities of vanishing and dichotomy cases of the minimizing sequence in the Nehari manifold. This argument is much simpler than the one used in dimension two where we consider the nonlinear Schrödinger equation -Δ⁢u+V⁢u=f⁢(u){-\Delta u+Vu=f(u)} with a degenerate potential V in ℝ2{\mathbb{R}^{2}}.

This paper analyzes the behavior of the positive solution θε{\theta_{\varepsilon}} of the perturbed problem { - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - [ a ε ⁢ ( x ) + b ε ⁢ ( x ) ] ⁢ u p = 0 in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-\Delta u}&\displaystyle=\lambda m(x)u-% [a_{\varepsilon}(x)+b_{\varepsilon}(x)]u^{p}=0&&\displaystyle\text{in}\ \Omega% ,\\ \displaystyle Bu&\displaystyle=0&&\displaystyle\text{on}\ \partial\Omega,\end{% aligned}\right. as ε↓0{\varepsilon\downarrow 0}, where aε⁢(x)≈εα⁢a⁢(x){a_{\varepsilon}(x)\approx\varepsilon^{\alpha}a(x)} and bε⁢(x)≈εβ⁢b⁢(x){b_{\varepsilon}(x)\approx\varepsilon^{\beta}b(x)} for some α≥0{\alpha\geq 0} and β≥0{\beta\geq 0}, and some Hölder continuous functions a⁢(x){a(x)} and b⁢(x){b(x)} such that a⪈0{a\gneq 0} (i.e., a≥0{a\geq 0} and a≢0{a\not\equiv 0}) and minΩ¯⁡b>0{\min_{\bar{\Omega}}b>0}. The most intriguing and interesting case arises when a⁢(x){a(x)} degenerates, in the sense that Ω0≡int⁡a-1⁢(0){\Omega_{0}\equiv\operatorname{int}a^{-1}(0)} is a non-empty smooth open subdomain of Ω, as in this case a “blow-up” phenomenon appears due to the spatial degeneracy of a⁢(x){a(x)} for sufficiently large λ. In all these cases, the asymptotic behavior of θε{\theta_{\varepsilon}} will be characterized according to the several admissible values of the parameters α and β. Our study reveals that there may exist two different blow-up speeds for θε{\theta_{\varepsilon}} in the degenerate case.

The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is - A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) ⁢ u ′′ ⁢ ( t ) = λ ⁢ f ⁢ ( t , u ⁢ ( t ) ) , t ∈ ( 0 , 1 ) , q ≥ 1 , -A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1, is considered. Due to the coefficient A⁢((b*uq)⁢(1)){A((b*u^{q})(1))} appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the (n-1,1){(n-1,1)}-conjugate problem.

This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data { - div ⁡ ( 𝐀 ⁢ ( x , D ⁢ u ) ) = g - div ⁡ f in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(\mathbf{A}(x,Du))&% \displaystyle=g-\operatorname{div}f&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. where Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}} is sufficiently flat in the sense of Reifenberg.

In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) ,  2 s ⋆ := 2 ⁢ N N - 2 ⁢ s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝN{\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D0s⁢(Ω){D^{s}_{0}(\Omega)} is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. us{u_{s}} converges to a l.e.s.s. ut{u_{t}} as s goes to any t>0{t>0}. In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t-ε{t-\varepsilon}. A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s>1{s>1}.

We study the asymptotic profile, as ℏ→0{\hbar\rightarrow 0}, of positive solutions to - ℏ 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u - ℏ 2 + γ ⁢ u ⁢ Δ ⁢ u 2 = K ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , x ∈ ℝ N , -\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2% }u,\quad x\in\mathbb{R}^{N}, where γ⩾0{\gamma\geqslant 0} is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L2{L^{2}}-energy solutions. We investigate the concentrating behavior of solutions when γ>0{\gamma>0} and, differently from the case γ=0{\gamma=0} where the leading potential is V , the concentration is here localized by the source potential K . Moreover, surprisingly for γ>0{\gamma>0} we find a different concentration behavior of solutions in the case p=2⁢NN-2{p=\frac{2N}{N-2}} and when 2⁢NN-2<p<4⁢NN-2{\frac{2N}{N-2}<p<\frac{4N}{N-2}}. This phenomenon does not occur when γ=0{\gamma=0}.

In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i ⁢ u t + Δ ⁢ u + | u | α ⁢ u - g ⁢ ( u t ) = 0   in  ⁢ Ω × ( 0 , ∞ ) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, n≤3{n\leq 3}, is a bounded domain with smooth boundary ∂⁡Ω=Γ{\partial\Omega=\Gamma} and α=2,3{\alpha=2,3}. Our goal is to consider a different approach than the one used in [B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z. 254 2006, 4, 729–749], so instead than using the properties of pseudo-differential operators introduced by cited authors, we consider a nonlinear damping, so that we remark that no growth assumptions on g⁢(z){g(z)} are made near the origin.

In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.

The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u→0u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R+n×R\mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R+n×R\mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space Rn-1×R\mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.