Band 20, Heft 4

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval I , under Dirichlet conditions in the exterior of I . This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov–Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.

In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation - u t ⁢ det ⁡ D 2 ⁢ u = f ⁢ ( x , t ) -u_{t}\det D^{2}u=f(x,t) and obtain the existence and uniqueness of viscosity solutions with asymptotic behavior by using the Perron method.

We consider energies on a periodic set ℒ{\mathcal{L}} of the form ∑i,j∈ℒai⁢jε⁢|ui-uj|{\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert}, defined on spin functions ui∈{0,1}{u_{i}\in\{0,1\}}, and we suppose that the typical range of the interactions is Rε{R_{\varepsilon}} with Rε→+∞{R_{\varepsilon}\to+\infty}, i.e., if |i-j|≤Rε{\lvert i-j\rvert\leq R_{\varepsilon}}, then ai⁢jε≥c>0{a^{\varepsilon}_{ij}\geq c>0}. In a discrete-to-continuum analysis, we prove that the overall behavior as ε→0{\varepsilon\to 0} of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on ε⁢ℒ{\varepsilon\mathcal{L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded Rε{R_{\varepsilon}} and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ℒ=ℤd{\mathcal{L}=\mathbb{Z}^{d}}.

The chemotaxis-growth system ($\star$) { u t = D ⁢ Δ ⁢ u - χ ⁢ ∇ ⋅ ( u ⁢ ∇ ⁡ v ) + ρ ⁢ u - μ ⁢ u α , v t = d ⁢ Δ ⁢ v - κ ⁢ v + λ ⁢ u {}\left\{\begin{aligned} \displaystyle{}u_{t}&\displaystyle=D\Delta u-\chi% \nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&\displaystyle=d\Delta v-\kappa v+\lambda u\end{aligned}\right. is considered under homogeneous Neumann boundary conditions in smoothly bounded domains Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, n≥1{n\geq 1}. For any choice of α>1{\alpha>1}, the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ($\star$), the present work shows that, whenever α≥2-2n{\alpha\geq 2-\frac{2}{n}}, under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state ((ρμ)1α-1,λκ⁢(ρμ)1α-1){\bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{% \kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)}} in the large time limit.

We study the existence of positive ground state solution for Choquard systems. In the autonomous case, we prove the existence of at least one positive ground state solution by the Pohozaev manifold method and symmetric-decreasing rearrangement arguments. Moreover, we show that each positive ground state solution is radial symmetric. While, in the nonautonomous case, a positive ground state solution is obtained by using a monotonicity trick and a global compactness lemma. We remark that, under our assumptions of the nonlinearity Wu{W_{u}}, the search of ground state solutions cannot be reduced to the study of critical points of a functional restricted to a Nehari manifold.

We consider some cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada–Kawasaki–Teramoto (SKT) model in population biology. We establish the global existence of strong solutions to systems for multiple species having equal either diffusion or reaction rates. The systems are given on bounded domains of arbitrary dimension.

In this paper, we consider a class of critical concave convex Ambrosetti–Prodi type problems involving the fractional p -Laplacian operator. By applying the linking theorem and the mountain pass theorem as well, the interaction of the nonlinearities with the first eigenvalue of the fractional p -Laplacian will be used to prove existence of multiple solutions.

We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight: ∂ t ⁡ u = ∂ x ⁢ x ⁡ ( u m ) + | x | σ ⁢ u p , \partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p}, posed for x∈ℝ{x\in\mathbb{R}}, t≥0{t\geq 0}, where m>1{m>1}, 0<p<1{0<p<1} and σ>2⁢(1-p)m-1{\sigma>\frac{2(1-p)}{m-1}}. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for m+p>2{m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when p<1{p<1}. We moreover prove that, if the condition m+p>2{m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity . We also show that there is no self-similar solution if m+p<2{m+p<2}, while the critical range m+p=2{m+p=2} with σ>2{\sigma>2} is postponed to a different work due to significant technical differences.

In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = | u | p - 2 ⁢ u in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-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 and bounded domain in ℝN,N≥3{\mathbb{R}^{N},N\geq 3}. More specifically we prove that, for p near the critical exponent 22*=4⁢N/(N-2){22^{*}=4N/(N-2)}, the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.