http://math.ucv.ro/~radulescu/articles/DGR2006.pdf

Lane-Emden-Fowler equations with convection and singular potential

On the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearities

We are concerned with singular elliptic problems of the form $-\Delta u\pm p(d(x))g(u)=\la f(x,u)+\mu |\nabla u|^a$ in $\Omega,$ where $\Omega$ is a smooth bounded domain in $\RR^N$, $d(x)={\rm dist}(x,\partial\Omega),$ $\la>0,$ $\mu\in\RR$, $0<a\leq 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive weight with possible singular behavior on the boundary of $\Omega$ and that the nonlinearity $g$ is unbounded around the origin. Taking into account the competition between the anisotropic potential $p(d(x))$, the convection term $|\nabla u|^a$, and the singular nonlinearity $g$, we establish various existence and nonexistence results.

Singular elliptic problems with convection term in anisotropic media

We study the weighted p-Laplacian with an isolated singular point, and discuss some properties of solutions near such a point, including removable singularities, boundedness and growth rate estimates. In particular, some relevant critical exponents are exactly determined.

Some properties of weak solutions to the weighted p-Laplacian near an isolated singular point

We show the existence of principal eigenvalues of the problem -Δu = Agu in R d where g is an indefinite weight function. The existence of a continuous family of principal eigenvalues is demonstrated. Also, we prove the existence of a principal eigenvalue for which the principal eigenfunction u → 0 at ∞.

/pdf/principal-eigenvalues-for-indefinite-weight-problems-in-all-4dgm2amrmt.pdf

Principal eigenvalues for indefinite weight problems in all of u^d

We prove the existence of positive eigenvalues for two types of equations having an indefinite weight. We consider the linear case, and we show the existence of a positive principal eigenvalue corresponding to a positive eigenfunction. For the nonlinear case, we show that the set of positive eigenvalues is an unbounded interval, and that the corresponding eigenfunctions are positive.

On the Existence of Positive Eigenvalues for Linear and Nonlinear Equations with Indefinite Weight

In this work, we study the existence of eigenvalues for a type of non-linear equations and we give some conditions to get positive eigenfunctions. Particularly, we show that under some appropriate conditions, the set of eigenvalues is an interval.

Eigenvalues in Non-Linear Potential Theory

In this paper, we prove the Harnack inequality for nonnegative weak solutions of the following nonlinear subelliptic equation −divA(x,u,∇u) = f(x,u,∇u).

Harnack inequality for nonlinear weighted equations

In this paper, we consider a problem of the form 

$$\left\{ \begin{gathered} \Delta {\kern 1pt} u\left( x \right) + {\lambda g}\left( x \right)f\left( {u\left( x \right)} \right) = 0\quad x \in \mathbb{R}^d , \hfill \\ \lim _{\left\| x \right\| \to \infty } u\left( x \right) = 0, \hfill \\ \end{gathered} \right.$$

where d≥3, f is a positive locally Lipschitz bounded function and g is assumed to change sign. We give some conditions of integral type to get the existence of positive solutions for λ large enough.

M. Mosbah

Papers

On the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearities

On the Existence of Positive Eigenvalues for Linear and Nonlinear Equations with Indefinite Weight

Eigenvalues in Non-Linear Potential Theory

Harnack inequality for nonlinear weighted equations

On the Existence of Positive Eigenvalues for Semilinear Elliptic Equation on all of R d