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Existence of strictly positive solutions for sublinear elliptic problems in bounded domains  [PDF]
Tomas Godoy,Uriel Kaufmann
Mathematics , 2013,
Abstract: Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}\xi^{p}\leq f\left(\xi\right) \leq k_{2}\xi^{p}$ for all $\xi\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right) f\left(u\right) $ in $\Omega$, $u=0$ on $\partial\Omega$.
Strictly positive solutions for one-dimensional nonlinear problems involving the p-Laplacian  [PDF]
Uriel Kaufmann,Ivan Medri
Mathematics , 2013, DOI: 10.1017/S0004972713000725
Abstract: Let $\Omega$ be a bounded open interval, and let $p>1$ and $q\in\left(0,p-1\right) $. Let $m\in L^{p^{\prime}}\left(\Omega\right) $ and $0\leq c\in L^{\infty}\left(\Omega\right) $. We study existence of strictly positive solutions for elliptic problems of the form $-\left(\left\| u^{\prime}\right\|^{p-2}u^{\prime}\right) ^{\prime}+c\left(x\right) u^{p-1}=m\left(x\right) u^{q}$ in $\Omega$, $u=0$ on $\partial\Omega$. We mention that our results are new even in the case $c\equiv0$.
Existence of positive solutions for some nonlinear elliptic problems in unbounded domains of  [PDF]
Noureddine Zeddini
Abstract and Applied Analysis , 2006, DOI: 10.1155/aaa/2006/24307
Abstract: This paper deals with a class of nonlinear elliptic equations inan unbounded domain D of ℝn, n≥3, with a nonempty compact boundary, where the nonlinear term satisfies someappropriate conditions related to a certain Kato classK∞(D). Our purpose is to give some existence results andasymptotic behaviour for positive solutions by using the Greenfunction approach and the Schauder fixed point theorem.
Existence of positive solutions for some nonlinear elliptic problems in unbounded domains of n
Noureddine Zeddini
Abstract and Applied Analysis , 2006,
Abstract: This paper deals with a class of nonlinear elliptic equations in an unbounded domain D of n , n≥3 , with a nonempty compact boundary, where the nonlinear term satisfies some appropriate conditions related to a certain Kato class K ∞ ( D ) . Our purpose is to give some existence results and asymptotic behaviour for positive solutions by using the Green function approach and the Schauder fixed point theorem.
Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components
Luisa Toscano,Speranza Toscano
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/760854
Abstract: We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.
Infinite multiplicity of positive solutions for singular nonlinear elliptic equations with convection term and related supercritical problems
Carlos C. Aranda
Electronic Journal of Differential Equations , 2009,
Abstract: In this article, we consider the singular nonlinear elliptic problem $$displaylines{ -Delta u = g(u)+h( abla u)+f(u) quadhbox{in }Omega, cr u = 0 quadhbox{on }partialOmega. }$$ Under suitable assumptions on $g, h, f, Omega$ that allow a singularity of g at the origin, we obtain infinite multiplicity results. Moreover, we state infinite multiplicity results for related boundary blow up supercritical problems and for supercritical elliptic problems with Dirichlet boundary condition.
Asymptotic Behavior in Degenerate Parabolic Fully Nonlinear equations and its application to Elliptic Eigenvalue Problems  [PDF]
Soojung Kim,Ki-ahm Lee
Mathematics , 2012,
Abstract: We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+ \mu\vp^{p}=0, \quad \vp>0\quad\text{in $\Omega$} \vp&=0\quad\text{on $\p\Omega$} posed in a (strictly) convex and smooth domain $\Omega\subset\re^n$ for $0< p \leq 1,$ where $F(\cdot)$ is uniformly elliptic, positively homogeneous of order one and concave. We establish that $\log (\vp)$ is concave in the case $p=1$ and that the function $\vp^{\frac{1-p}{2}}$ is concave for $0
Multiple Positive Solutions for a Class of Nonlinear Elliptic Eigenvalue Problems with a Sign-Changing Nonlinearity  [PDF]
M. L. Carvalho,J. V. Goncalves,K. O. Silva
Mathematics , 2013,
Abstract: In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions for existence were also established. In the present paper the authors extend the main results by Loc and Schmitt to the $\Phi$-Laplacian. To overcome the difficulties with this much more general operator it was necessary to employ regularity results by Lieberman, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions by Le.
Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems  [PDF]
Teresa D'Aprile,David Ruiz
Mathematics , 2009,
Abstract: We study the existence of positive and sign-changing multipeak solutions for the stationary Nonlinear Schroedinger Equation. Here no symmetry on $V$ is assumed. It is known that this equation has positive multipeak solutions with all peaks approaching a local maximum of the potential. It is also proved that solutions alternating positive and negative spikes exist in the case of a minima. The aim of this paper is to show the existence of both positive and sign-changing multipeak solutions around a nondegenerate saddle point of the external potential.
Existence theorems for nonlinear elliptic problems
Halidias Nikolaos
Journal of Inequalities and Applications , 2001,
Abstract: In this paper we prove two theorems for noncoercive elliptic boundary value problems using the critical point theory of Chang and the subdifferentiable of Clarke. The first result is for a Dirichlet noncoercive problem and the second one is for Neumann elliptic problem with nonlinear multivalued boundary conditions. We use the mountain-pass and the saddle-point theorems to obtain nontrivial solutions for these problems.
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