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In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems: \begin{equation*} \left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda|u|^{p - 2}u+ \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \;\;\; \mbox{in}\;\;\; \mathbb{R}^N, (-\Delta)^sv +V_2(x)v= \lambda|v|^{q - 2}v+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \;\;\; \mbox{in}\;\;\; \mathbb{R}^N, (u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N). \end{array}\right. \end{equation*} Here we mention that $\alpha > 1, \beta > 1, 1 \leq p \leq q < 2 < \alpha + \beta < 2^*_s$, $\theta > 0, \lambda > 0, N > 2s$, and $s \in (0,1)$. Notice also that continuous potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ satisfy some extra assumptions. Furthermore, we find the largest positive number $\lambda^* > 0$ such that our main problem admits at least two positive solutions for each $ \lambda \in (0, \lambda^*)$. This can be done by using the nonlinear Rayleigh quotient together with the Nehari method. The main feature here is to minimize the energy functional in Nehari manifold which allows us to prove our main results without any restriction on size of parameter $\theta > 0$., Comment: In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems

In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space $\mathbb{R}^N$. More precisely, we consider the following nonlocal elliptic problem: \begin{equation*} - \Delta u + V(x)u = \lambda a(x) |u|^{q-2} u + \displaystyle \int \limits_{\mathbb{R}^N}\frac{b(y)\vert u(y) \vert^p dy}{\vert x\vert^\alpha\vert x-y\vert^\mu \vert y\vert^\alpha} b(x)\vert u\vert^{p-2}u, \,\, \hbox{in}\ \mathbb{R}^N, \,\, u\in H^1(\mathbb{R}^N), \end{equation*} where $\lambda >0, \alpha \in (0,N), N\geq3, 0<\mu 0$ in $\mathbb{R}^N$ and $b\in{L}^{t}(\mathbb{R}^N), b>0$ in $\mathbb{R}^N$ for some specific $r, t > 1$. We assume also that $1\leq q<2$ and $2_{\alpha,\mu} < p<2_{\alpha,\mu}^*$ where $2_{\alpha ,\mu}=(2N-2\alpha-\mu)/N$ and $2_{\alpha,\mu}^*= (2N-2\alpha-\mu)/(N-2)$. Our main contribution is to find the largest $\lambda^* > 0$ in such way that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. In order to do that we apply the nonlinear Rayleigh quotient together with the Nehari method. Moreover, we prove a Brezis-Lieb type Lemma and a regularity result taking into account our setting due to the potentials $a, b : \mathbb{R}^N \to \mathbb{R}$., Comment: In the present work, we consider existence and multiplicity of positive solutions for nonlocal elliptic problems driven by the Stein-Weiss problem with concave-convex nonlinearities defined in the whole space $\mathbb{R}^N$

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$ \left\{ \begin{array}{lr} -\Delta_\Phi u = \lambda a(x) |u|^{q-2}u + |u|^{p-2}u, & x\in\Omega, u = 0, & x \in \partial \Omega, \end{array} \right. $$ where $\Omega \subset \mathbb{R}^N, N \geq 2,$ is a smooth bounded domain, $1 < q < \ell \leq m < p < \ell^*$ and $\Phi: \mathbb{R} \to \mathbb{R}$ is suitable $N$-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number $\lambda^* > 0$ in such way that our main problem admits at least two distinct solutions for each $\lambda \in (0, \lambda^*)$. Furthermore, using some fine estimates and some extra assumptions on $\Phi$, we prove the existence of at least two positive solutions for $\lambda = \lambda^*$ and $\lambda \in (\lambda^*, \overline{\lambda})$ where $\overline{\lambda} > \lambda^*$.

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger equation $(-\Delta )^s u + V(x) u = \lambda a(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $00,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp $\lambda^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $\lambda \in (0, \lambda^*)$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered.

A minimax variational principle for saddle-point solutions with prescribed energy levels is introduced. The approach is based on the development of the linking theorem to the energy level nonlinear generalized Rayleigh quotients. An application to indefinite elliptic Dirichlet problems is presented. Among the consequences, the existence of solutions with zero-energy levels is obtained., Comment: 9 pages

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -\Delta u +V(x) u &=& (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array} $$ where $\lambda > 0, N \geq 3, \alpha \in (0, N)$. The potential $V$ is a continuous function and $I_\alpha$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_{\alpha} < p < 2^*_\alpha$ where $2_\alpha=(N+\alpha)/N$, $2_\alpha=(N+\alpha)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $\lambda > 0$ taking into account the nonlinear Rayleigh quotient. More precisely, there exists $\lambda_n > 0$ such that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda_n]$. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $\lambda_n > 0$ is optimal in some sense which allow us to apply the Nehari method.

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' "vanishing" Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schr\"{o}dinger equations, taking into account unbounded or bounded potentials.

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (\phi 1, \phi 2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases., Comment: Here we discusse existence and multiplicity of solutions for for quasilinear elliptic systems driven by (\phi 1, \phi 2)-Laplacian operator

In this paper we consider the nonlinear Choquard equation $$ -\Delta u+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<\mu

It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by $\Phi$-Laplacian operator. Here we consider the reflexive and nonreflexive cases using an auxiliary problem. In order to prove our main results we employ variational methods, regularity results and truncation techniques.

\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -\Delta_{\Phi}{u}=f(u), \quad \mbox{in} \quad \Omega u=0, \quad \mbox{on} \quad \partial \Omega, \end{array} \right. $$ where $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a smooth bounded domain, $f:\mathbb{R} \to \mathbb{R}$ is a continuous function verifying some conditions, and $\Phi:\mathbb{R} \to \mathbb{R}$ is a N-function which is not assumed to satisfy the well known $\Delta_2$-condition, then the Orlicz-Sobolev space $W^{1,\Phi}_0(\Omega)$ can be non reflexive. As main model we have the function $\Phi(t)=(e^{t^{2}}-1)/2$. Here, we study some situations where it is possible to work with global minimization, local minimization and mountain pass theorem, however some estimates are not standard for this type of problem., Comment: In this new version, we improve the Theorems 1.2 and 1.3, in the sense that we remove the assumption $2diam(Omega)\leq 1$

It is established existence and multiplicity of solutions for strongly nonlinear problems driven by the $\Phi$-Laplacian operator on bounded domains. Our main results are stated without the so called $\Delta_{2}$ condition at infinity which means that the underlying Orlicz-Sobolev spaces are not reflexive.

In this work we establish existence and multiplicity of solutions for elliptic problem with nonlinear boundary conditions under strong resonance conditions at infinity. The nonlinearity is resonance at infinity and the reso- nance phenomena occurs precisely in the first Steklov eigenvalue problem. In all results we use Variational Methods, Critical Groups and the Morse Theory.

In this paper we establish existence and multiplicity of solutions for an elliptic system which has strong resonance at first eigenvalue. To describe the resonance, we use an eigenvalue problem with indefinite weight. In all results we use Variational Methods.

We establish existence and multiplicity of solutions for a elliptic resonant elliptic problem under Dirichlet boundary conditions., Comment: This is a research to resonant elliptic problems under Cerami condition using variational methods