arXiv subjects

Emer Lopera

Publications and source records attributed to Emer Lopera.

5 recordsLinked to original sources

arXiv2025

Nonexistence of positive radial solutions for semipositone $\phi$-Laplacian problems with superlinear reaction term

The aim of this paper is to prove the nonexistence of positive radial solutions to the problem $-\Delta_\phi u=\lambda f(u)$, $x\in B_1(0)$, $u(x)=0$ on $|x|=1$, for $\lambda>0$ sufficiently large. Here, $\phi$ is a continuous function, $\Delta_\phi$ denotes the $\phi$-Laplacian operator which is defined by $\Delta_\phi (u):=div (\phi (|\nabla u|) \nabla u)$, and $B_1(0)$ is the unit ball in $\mathbb{R}^N$, with $N>1$. Furthermore, $f$ is a continuous, nondecreasing function such that $f(0)<0$, and its behavior at infinity is intimately related to $\phi$. Our findings are presented in a combined format, employing both an indirect argument and an energy analysis.

math.AP↗

arXiv2025

Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations

We study the solvability of $(p,q)$-Laplacian problems with nonlinear reaction terms and non-homogeneous Neumann boundary conditions. First, we provide a complete description of the spectrum of the eigenvalue problem involving the $(p,q)$-Laplacian with weights and a spectral parameter present in both the differential equation and on the boundary. Then, using variational methods and critical point theory, we prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically, in some sense, below the first eigenvalue of the $q$-Laplacian problem with weights and a spectral parameter present in both the differential equation and on the boundary. We also establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below the first Steklov-Neumann eigenvalue-line, which is a line connecting the first Steklov and first Neumann eigenvalues for $q$-Laplacian problems with weights and a spectral parameter present either in the differential equation or on the boundary.

math.AP↗

arXiv2025

Multiplicity results for non-local operators of elliptic type

In this paper, we study a class of problems proposed by Servadei and Valdinoci in \cite{Ser3}; namely, \begin{equation}\label{prob_0} \left\{\begin{aligned} -\mathcal{L}_{K} u(x)-\lambda u(x) & =f(x,u), \mbox{ for } x\in \Omega; u & =0 \quad \text{ in } \mathbb{R}^{N}\backslash\Omega, \end{aligned} \right. \end{equation} where $\Omega\subset \mathbb{R}^{N}$ is an open bounded set with Lipschitz boundary, $\lambda\in\mathbb{R}$, $f\in C^{1}(\overline{\Omega}\times\mathbb{R},\mathbb{R})$, with $f(x,0) = 0$ for $x\in\Omega$, and $\mathcal{L}_K$ is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem (1) at the origin and at infinity, respectively, we prove that problem (\ref{prob_0}) has three nontrivial solutions for the case $\lambda < \lambda_1$ and two nontrivial solutions for the case $\lambda\geqslant\lambda_1,$ where $\lambda_1$ is the first eigenvalue of the operator $-\mathcal{L}_K$. Finally, assuming that the nonlinearity $f$ is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem (1) for the case $\lambda\geqslant\lambda_1$. We use variational methods and infinite-dimensional Morse theory to obtain the results.

math.AP↗

arXiv2024

Multiplicity results for Schr\"odinger type fractional $p$-Laplacian boundary value problems

In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(\Delta)_{p}^{s} u + V(x)|u|^{p-2}u &= \lambda f(u),&x\in\Omega; u&=0,&x\in \R^{N}\backslash\Omega, \end{aligned} \right. \end{equation} where $\Omega\subset\R^{N}$ is an open bounded set with Lipschitz boundary $\partial\Omega$, $N\geqslant 2,$ $V\in L^{\infty}(\R^{N})$, and $(-\Delta)_p^s$ denotes the fractional $p$-Laplacian with $s\in(0,1), 1 0$, and $f:\R\rightarrow\R$ is a continuous function. We extend the results of Lopera {\it et al.} in \cite{Lopera1} by proving the existence of a second weak solution for problem (\ref{probaa1}). We apply a variant of the mountain-pass theorem due to Hofer \cite{Hofer2} and infinite-dimensional Morse theory to obtain the existence of at least two solutions.

math.AP↗

arXiv2022

Existence of positive solutions for a parameter fractional $p$-Laplacian problem with semipositone nonlinearity

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-\Delta)_p^s(u) &=& \lambda f(u) \qquad & \text{in} \ \ \Omega \\u &=& 0 & \text{in} \ \ \mathbb{R}^N -\Omega , \end{array}\right. \] whenever $\lambda >0$ is a sufficiently small parameter. Here $\Omega \subseteq \mathbb{R}^N$ a bounded domain with $C^{1,1}$ boundary, $2\leqslant p 0$ is chosen sufficiently small the associated Energy Functional to the problem has a mountain pass structure and, therefore, it has a critical point $u_\lambda$, which is a weak solution. After that we manage to prove that this solution is positive by using new regularity results up to the boundary and a Hopf's Lemma.

math.AP↗