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It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: \[ \left\{\begin{array}{@{}l} -\Delta u +V(x)u = \mu (I_{\alpha_1}*|u|^q)|u|^{q-2}u-\lambda (I_{\alpha_2}*|u|^p)|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^N, \\ u\in H^1(\mathbb{R}^N), \end{array}\right. \] { − Δ u + V (x) u = μ (I α 1 ∗ | u | q) | u | q − 2 u − λ (I α 2 ∗ | u | p) | u | p − 2 u in R N , u ∈ H 1 (R N) , where $ p \gt q, \lambda, \mu \gt 0 $ p > q , λ , μ > 0 , $ \alpha _1\leq \alpha _2 $ α 1 ≤ α 2 ; $ \alpha _1,\alpha _2\in (0,N), N \geq 3 $ α 1 , α 2 ∈ (0 , N) , N ≥ 3 ; $ p\in (2_{\alpha _2},2^*_{\alpha _2}) $ p ∈ (2 α 2 , 2 α 2 ∗) ; $ q\in (2_{\alpha _1},2^*_{\alpha _1}) $ q ∈ (2 α 1 , 2 α 1 ∗) , $ 2_{\alpha _j}=(N+\alpha _j)/N $ 2 α j = (N + α j) / N and $ 2^*_{\alpha _j}=(N+\alpha _j)/(N-2), j =1, 2 $ 2 α j ∗ = (N + α j) / (N − 2) , j = 1 , 2. Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp $ \mu _n \gt 0 $ μ n > 0 and $ \lambda ^*, \lambda _* \gt 0 $ λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each $ \mu \gt \mu _n $ μ > μ n where $ \lambda \in (0, \min (\lambda ^*, \lambda _*)) $ λ ∈ (0 , min (λ ∗ , λ ∗)). The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each $ \mu \lt \mu _n $ μ < μ n and $ \lambda \gt 0 $ λ > 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Choquard equation $$-\Delta u + V(x)u = \left({\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$ where 0 < μ < N , N ⩾ 3, g (u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and $G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potential V (x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma. [ABSTRACT FROM AUTHOR]

We establish the existence and multiplicity of solutions for a class of fourth-order superlinear elliptic problems under Navier conditions on the boundary. Here we do not use the Ambrosetti-Rabinowitz condition; instead we assume that the nonlinear term is a nonlinear function which is nonquadratic at infinity. [ABSTRACT FROM AUTHOR]

We establish the existence of nontrivial solutions for an elliptic system which is resonant both at the origin and at infinity. The resonance is given by an eigenvalue problem with indefinite weight, and the nonlinear term is permitted to be unbounded. Also, we consider the case where the resonance at infinity and at the origin can occur with different weights. Our main tool is the computation of critical groups. [ABSTRACT FROM AUTHOR]