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 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 $\partial\Omega$.
 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 $c\equiv0$.
 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.
 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 $0  Mathematics , 2013, Abstract: In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions for existence were also established. In the present paper the authors extend the main results by Loc and Schmitt to the$\Phi$-Laplacian. To overcome the difficulties with this much more general operator it was necessary to employ regularity results by Lieberman, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions by Le.  Mathematics , 2009, Abstract: We study the existence of positive and sign-changing multipeak solutions for the stationary Nonlinear Schroedinger Equation. Here no symmetry on$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.