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By means of the variational approximation (VA) and systematic simulations, we study dynamics and stability boundaries for fundamental solitons in a two-dimensional (2D) self-attracting Bose-Einstein condensate (BEC), trapped in an quasiperiodic optical lattice (OL), with the amplitude subject to periodic time modulation (the modulation frequency, ω. Regions of stability of the solitons against the collapse and decay are identified in the space of the model’s parameters.

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter $\lambda$ under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted $p$-Laplacian operator and subcritical nonlinearities, and even in the case $p=2$ the main existence results are new. Denoting by $\lambda_1$ the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case $\lambda\ge \lambda_1$ or $\lambda

In this paper we are interested in the multiplicity of weak solutions to the following Neumann problem involving the $p(x)$-Laplacian operator $$ \left\{ \begin{array}{ll} -\delta_{p(x)}u + \mid u \mid^{p(x)-2}u = \lambda \alpha(x) f(u) + \beta(x) g(u) \ \ \ & in \ \Omega \\ \frac{\partial u}{\partial v} = 0 \ \ \ & on \ \Omega\end{array} \right. $$ We establish the existence of at least three solutions to this problem by using, as main tool, a recent variational principle due to Ricceri.

By using a recent result of Ricceri, we prove the existence of multiple solutions for perturbed $n$-Laplacian equations with Dirichlet boundary conditions. The Trudinger Moser inequality allows us to deal with perturbations with exponential growth.

In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain $\Omega$, which is the $p$--Laplacian equation, $-\mbox{div}(|\nabla u|^{p-2} \nabla u) =f$, in a subdomain $\Omega_1$ of $\Omega$ and the Laplace equation, $-\Delta u = f$, in its complementary (that is, our equation involves the so-called $p(x)$--Laplacian with a discontinuous exponent). We assume that the right-hand side $f$ belongs to $L^\infty(\Omega)$. For this problem, we study the behaviour of the solutions as $p$ goes to $1$, showing that they converge to a function $u$, which is almost everywhere finite when the size of the datum $f$ is small enough. Moreover, we prove that this $u$ is a solution of a limit problem involving the $1$-Laplacian operator in $\Omega_1$. We also discuss uniqueness under a favorable geometry.

We consider the following quasilinear elliptic system: \begin{equation} \begin{cases} \Delta u+u(a-u-cv)=0 \ \ & \mbox{in} \ \ \Omega ,\\ \Delta \Big [ \Big ( 1+\dfrac{\gamma }{1+\beta u} \Big ) v \Big ]+v(b+du-v)=0 \ \ & \mbox{in} \ \ \Omega ,\\ u=v=0 \ \ & \mbox{on} \ \ \partial\Omega, \end{cases} \nonumber \end{equation} where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$. This system is a stationary problem of a prey-predator model with non-linear diffusion $\Delta (\frac{v}{1+\beta u})$, and $u$ (respectively $v$) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large $\beta $ under the restriction $b>(1+\gamma )\lambda_{1}$, where $\lambda_{1}$ is the least eigenvalue of $-\Delta$ with homogeneous Dirichlet boundary condition. The present paper studies two {\it shadow systems} and gives the complete limiting characterization of positive solutions as $\beta\to\infty$ without any restriction on $b$.