arXiv subjects

Humberto Ramos Quoirin

Publications and source records attributed to Humberto Ramos Quoirin.

27 records · Page 2Linked to original sources

arXiv2017

Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:Ω\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_Ωa<0$. We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $\frac{\partial u}{\partialν}=0$ on $\partialΩ$, if $q_{0} 0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $Ω$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ ensuring the existence of a positive solution of $(P_{a,q})$. We also obtain some properties of the set of $q$'s such that $(P_{a,q})$ admits a solution which is positive on $\overlineΩ$. Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method. Several methods and results apply as well to the Dirichlet counterpart of $(P_{a,q})$.

math.AP↗

arXiv2017

An indefinite concave-convex equation under a Neumann boundary condition II

We proceed with the investigation of the problem $(P_λ): $ $-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } Ω, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial Ω$, where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_λ)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method.

math.AP↗

arXiv2017

Positive solutions of indefinite semipositone problems via sub-super solutions

Let $Ω\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:Ω\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-Δu=λm(x)(f(u)-k)$ in $Ω$, $u=0$ on $\partialΩ$, where $λ,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $λ$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.

math.AP↗

arXiv2017

Elliptic equations involving the $p$-Laplacian and a gradient term having natural growth

We investigate the problem $$ \left\{ \begin{array}{ll} -Δ_p u = g(u)|\nabla u|^p + f(x,u) \ & \mbox{in} \ \ Ω, \ \ \\ u>0 \ &\mbox{in} \ \ Ω, \ \ u = 0 \ &\mbox{on} \ \ \partialΩ, \end{array} \right. \leqno{(P)} $$ in a bounded smooth domain $Ω\subset \mathbb{R}^N$. Using a Kazdan-Kramer change of variable we reduce this problem to a quasilinear one without gradient term and therefore approachable by variational methods. In this way we come to some new and interesting problems for quasilinear elliptic equations which are motivated by the need to solve $(P)$. Among other results, we investigate the validity of the Ambrosetti-Rabinowitz condition according to the behavior of $g$ and $f$. Existence and multiplicity results for $(P)$ are established in several situations.

math.AP↗

arXiv2016

Positivity results for indefinite sublinear elliptic problems via a continuity argument

We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some positivity results for indefinite concave-convex type problems.

math.AP↗

arXiv2016

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem $$-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } Ω, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial Ω, \leqno{(P_λ)} $$ where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|λ|$ small. We characterize then the asymptotic profiles of these solutions as $λ\to 0$, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of $(P_λ)$.

math.AP↗

arXiv2015

Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity, stability and asymptotic profiles

We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter $λ$, which is inversely proportional to the diffusion rate. We establish several multiplicity results when the diffusion rate is large and analyze the asymptotic profiles and the stability properties of these steady states as the diffusion rate grows to infinity. In particular, our results show that in some cases bifurcation from zero and from infinity occur at $λ=0$. Our approach combines variational and bifurcation techniques.

math.AP↗

arXiv2015

Ground states of elliptic problems involving non homogeneous operators

We investigate the existence of ground states for functionals with nonhomogenous principal part. Roughly speaking, we show that the Nehari manifold method requires no homogeinity on the principal part of a functional. This result is motivated by some elliptic problems involving nonhomogeneous operators. As an application, we prove the existence of a ground state and infinitely many solutions for three classes of boundary value problems.

math.AP↗

arXiv2014

Multiple solutions for an indefinite elliptic problem with critical growth in the gradient

We consider the problem $(P)$, $$ -Δu =c(x)u+μ|\nabla u|^2 +f(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω),$$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $μ>0, \, c \in \mathcal{C}(\overlineΩ),$ and $ f \in L^q(Ω)$ for some $ q>\frac{N}{2}$ with $ f\gneqq 0. $ Here $c$ is allowed to change sign. We show that when $c^+ \not \equiv 0$ and $c^+ +μf$ is suitably small, this problem has at least two positive solutions. This result contrasts with the case $c \leq 0$, where uniqueness holds. To show this multiplicity result we first transform $(P)$ into a semilinear problem having a variational structure. Then we are led to the search of two critical points for a functional whose superquadratic part is indefinite in sign and has a so called slow growth at infinity. The key point is to show that the Palais-Smale condition holds.

math.AP↗