arXiv subjects

Edcarlos Domingos da Silva

Publications and source records attributed to Edcarlos Domingos da Silva.

3 recordsLinked to original sources

arXiv2025

Classical Sobolev approach for a critical fourth-order Leray-Lions type problem: existence and multiplicity of solutions

A fourth-order elliptic problem of Leray-Lions type is considered for combined nonlinearities and Sobolev-critical growth with Navier and Dirichlet boundary conditions. By combining variational methods and critical point theory, the existence and multiplicity of weak solutions are established in the setting of classical Sobolev spaces. Two distinct asymptotic regimes are considered for the perturbation term: sublinear and superlinear. In the sublinear case, the existence of infinitely many solutions is proved by using topological tools such as Krasnosel'skii's genus and Clark's deformation lemma. In the superlinear case, the existence of at least one nontrivial solution is obtained via the Mountain Pass Theorem. Furthermore, the applicability of the main results is illustrated in the context of Hamiltonian systems.

math.AP↗

arXiv2018

Existence of bound and ground states for fractional coupled systems in $\mathbb{R}^{N}$

In this work we consider the following class of nonlocal linearly coupled systems involving Schrödinger equations with fractional laplacian $$ \left\{ \begin{array}{lr} (-Δ)^{s_{1}} u+V_{1}(x)u=f_{1}(u)+λ(x)v, & x\in\mathbb{R}^{N}, (-Δ)^{s_{2}} v+V_{2}(x)v=f_{2}(v)+λ(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $(-Δ)^{s}$ denotes de fractional Laplacian, $s_{1},s_{2}\in(0,1)$ and $N\geq2$. The coupling function $λ:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is related with the potentials by $|λ(x)|\leq δ\sqrt{V_{1}(x)V_{2}(x)}$, for some $δ\in(0,1)$. We deal with periodic and asymptotically periodic bounded potentials. On the nonlinear terms $f_{1}$ and $f_{2}$, we assume "superlinear" at infinity and at the origin. We use a variational approach to obtain the existence of bound and ground states without assuming the well known Ambrosetti-Rabinowitz condition at infinity. Moreover, we give a description of the ground states when the coupling function goes to zero.

math.AP↗

arXiv2018

Positive ground states for a class of superlinear $(p,q)$-Laplacian coupled systems involving Schrödinger equations

We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{array}{lr} -Δ_{p} u+a(x)|u|^{p-2}u=f(u)+ αλ(x)|u|^{α-2}u|v|^β, & x\in\mathbb{R}^{N}, -Δ_{q} v+b(x)|v|^{q-2}v=g(v)+ βλ(x)|v|^{β-2}v|u|^α, & x\in\mathbb{R}^{N}, \end{array} \right. \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $λ(x)$ of the coupling term is related with the potentials by the condition $|λ(x)|\leqδa(x)^{α/p}b(x)^{β/q}$ where $δ\in(0,1)$ and $α/p+β/q=1$. We deal with periodic and asymptotically periodic potentials. The nonlinear terms $f(s), \; g(s)$ are "superlinear" at $0$ and at $\infty$ and are assumed without the well known Ambrosetti-Rabinowitz condition at infinity. Thus, we have established the existence of positive ground states solutions for a large class of nonlinear terms and potentials. Our approach is variational and based on minimization technique over the Nehari manifold.

math.AP↗