Published in Advanced Nonlinear Studies
We consider energies on a periodic set ℒ{\mathcal{L}} of the form ∑i,j∈ℒaijε|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{\l...
Published in Advanced Nonlinear Studies
The chemotaxis-growth system ($\star$){ut=DΔu-χ∇⋅(u∇v)+ρu-μuα,vt=dΔv-κv+λu{}\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 condition...
Published in Advanced Nonlinear Studies
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 s...
Published in Advanced Nonlinear Studies
In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: {-Δu-Δ(u2)u=|u|p-2uin Ω,u=0on ∂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 ...
Published in Advanced Nonlinear Studies
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 applicabil...
Published in Advanced Nonlinear Studies
We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight: ∂tu=∂xx(um)+|x|σup,\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 2(1-p)m-1{\sigma>\frac{2(1-p)}{m-1}}. As a fir...
Published in Advanced Nonlinear Studies
In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation -utdetD2u=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.
Published in Advanced Nonlinear Studies
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-dim...
Published in Advanced Nonlinear Studies
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 n...
Published in Advanced Nonlinear Studies
In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbe...
