Volume 19, Issue 1

This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, d⁢ℒ⁢u=u⁢h⁢(u,x){d\mathcal{L}u=uh(u,x)}, under non-classical mixed boundary conditions, ℬ⁢u=0{\mathcal{B}u=0} on ∂⁡Ω{\partial\Omega}. Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of ∂⁡Ω{\partial\Omega} through the regularity of the associated conormal projections and conormal distances . This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

In this paper, we obtain the existence and multiplicity of sign-changing solutions of the fractional p -Laplacian problems by applying the method of invariant sets of descending flow and minimax theory. In addition, we prove that the problem admits at least one least energy sign-changing solution by combining the Nehari manifold method with the constrained variational method and Brouwer degree theory. Furthermore, the least energy of sign-changing solutions is shown to exceed twice that of the least energy solutions.

The paper is related to a conjecture by Pegon, Santambrogio and Xia concerning the dimension of the boundary of some sets which we are calling “irrigation balls”. We propose a notion of sub-balls and sub-spheres of prescribed radius and we prove that, generically, the only possible Minkowski dimension of sub-spheres is the one expected in the conjecture. At the same time, beside the scale transition properties and the dimension estimates on some significant sets, we propose a third approach to study the fractal regularity which relies on lower oscillation estimates on the landscape function, which turns out to behave as a Weierstrass-type function.

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups), we prove two multiplicity theorems producing four and five, respectively, nontrivial smooth solutions when the parameter λ>0{\lambda>0} is small.

In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war u t - Δ ∞ β ⁢ u = f ⁢ ( x , t ) , {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and Δ∞β{\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator Δ ∞ β ⁢ u = Δ ∞ N ⁢ u + β ⁢ | D ⁢ u | \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x . Finally, when f=0{f=0}, we show some explicit solutions.

We consider the following class of fractional Schrödinger equations: ( - Δ ) α ⁢ u + V ⁢ ( x ) ⁢ u = K ⁢ ( x ) ⁢ f ⁢ ( u )   in  ⁢ ℝ N , (-\Delta)^{\alpha}u+V(x)u=K(x)f(u)\quad\text{in }\mathbb{R}^{N}, where α∈(0,1){\alpha\in(0,1)}, N>2⁢α{N>2\alpha}, (-Δ)α{(-\Delta)^{\alpha}} is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

This paper is concerned with the existence of a heteroclinic solution for the following class of elliptic equations: - Δ ⁢ u + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ′ ⁢ ( u ) = 0   in  ⁢ Ω , -\Delta{u}+A(\epsilon x,y)V^{\prime}(u)=0\quad\mbox{in }\Omega, where ϵ>0{\epsilon>0}, Ω=ℝ×𝒟{\Omega=\mathbb{R}\times\mathcal{D}} is an infinite cylinder of ℝN{\mathbb{R}^{N}} with N≥2{N\geq 2}. Here, we consider a large class of potentials V that includes the Ginzburg–Landau potential V⁢(t)=(t2-1)2{V(t)=(t^{2}-1)^{2}} and two geometric conditions on the function A . In the first condition we assume that A is asymptotic at infinity to a periodic function, while in the second one A satisfies 0 < A 0 = A ⁢ ( 0 , y ) = inf ( x , y ) ∈ Ω ⁡ A ⁢ ( x , y ) < lim inf | ( x , y ) | → + ∞ ⁡ A ⁢ ( x , y ) = A ∞ < ∞   for all  ⁢ y ∈ 𝒟 . 0<A_{0}=A(0,y)=\inf_{(x,y)\in\Omega}A(x,y)<\liminf_{|(x,y)|\to+\infty}A(x,y)=A% _{\infty}<\infty\quad\text{for all }y\in\mathcal{D}.

We study global continuation properties of the set of T -periodic solutions of parameterized second order delay differential equations with constant time lag on smooth manifolds. We apply our results to get multiplicity of T -periodic solutions. Our topological approach is mainly based on the notion of degree of a tangent vector field.

The purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake S⊆ℝ2{S\subseteq\mathbb{R}^{2}} this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s→1-{s\to 1^{-}} of the fractional perimeter of a set having locally finite (classical) perimeter.

In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in Ws,p⁢(ℝN){W^{s,p}(\mathbb{R}^{N})}. Define ∥ u ∥ 1 , τ = ( [ u ] W s , p ⁢ ( ℝ N ) p + τ ⁢ ∥ u ∥ p p ) 1 p   for any  ⁢ τ > 0 . \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds sup u ∈ W s , p ⁢ ( ℝ N ) , ∥ u ∥ 1 , τ ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) < + ∞ , \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where s∈(0,1){s\in(0,1)}, s⁢p=N{sp=N}, α∈[0,α*){\alpha\in[0,\alpha_{*})} and Φ N , s ⁢ ( t ) = e t - ∑ i = 0 j p - 2 t j j ! . \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: ( - Δ ) p s ⁢ u ⁢ ( x ) + V ⁢ ( x ) ⁢ | u ⁢ ( x ) | p - 2 ⁢ u ⁢ ( x ) = f ⁢ ( x , u ) + ε ⁢ h ⁢ ( x )   in  ⁢ ℝ N , (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where V⁢(x){V(x)} has a positive lower bound, f⁢(x,t){f(x,t)} behaves like eα⁢|t|N/(N-s){e^{\alpha\lvert t\rvert^{N/(N-s)}}}, h∈(Ws,p⁢(ℝN))*{h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and ε>0{\varepsilon>0}. Moreover, we also derive a weak solution with negative energy.

We study the following generalized quasilinear Schrödinger equation: - ( g 2 ⁢ ( u ) ⁢ ∇ ⁡ u ) + g ⁢ ( u ) ⁢ g ′ ⁢ ( u ) ⁢ | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ u = h ⁢ ( u ) , x ∈ ℝ N , -(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N}, where N≥3{N\geq 3}, g:ℝ→ℝ+{g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}} is an even differentiable function such that g′⁢(t)≥0{g^{\prime}(t)\geq 0} for all t≥0{t\geq 0}, h∈C1⁢(ℝ,ℝ){h\in C^{1}(\mathbb{R},\mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential V⁢(x):ℝN→ℝ{V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on 𝐑N{\mathbf{R}}^{N}, Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.