Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations

In this paper we consider the existence of antisymmetric solutions for the quasilinear defocusing Schrödinger equation in H1 (RN): −∆u + k 2 u∆u 2 + V(x)u = g(u), where N ≥ 3, V(x) is a positive continuous potential, g(u) is of subcritical growth and k is a non-negative parameter. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of antisymmetric solutions via a deformation lemma

Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity

AbstractWe are concerned with the following quasilinear elliptic equation$$\begin{array}{} \displaystyle -{\it\Delta} u-{\it\Delta}(u^{2})u=\mu |u|^{q-2}u+|u|^{2\cdot 2^*-2}u, u\in H_0^1({\it\Omega}), \end{array}$$(QSE)whereΩ⊂ ℝNis a bounded domain,N≥ 3,qN 0 such that (QSE) admits at least catΩ(Ω) positive solutions whenμ∈ (0,μ*)

Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations

In this paper we consider the existence of antisymmetric solutions for the quasilinear defocusing Schrödinger equation in $H^1(\mathbb{R}^N)$: $$-\Delta u +\frac{k}{2}u \Delta u^2+V(x)u=g(u),$$ where $N\geq 3$, $V(x)$ is a positive continuous potential, $g(u)$ is of subcritical growth and $k$ is a non-negative parameter. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of antisymmetric solutions via a deformation lemma

A multiplicity result for a (p, q)-Schrödinger–Kirchhoff type equation

none2noIn this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.openAmbrosio V.; Isernia T.Ambrosio, V.; Isernia, T

Standing waves with a critical frequency for nonlinear Choquard equations

In this paper, we study the nonlocal Choquard equation $$-\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon= (I_\alpha * |u_\varepsilon|^p)|u_\varepsilon|^{p-2}u_\varepsilon$$ where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$ and $\varepsilon>0$ is a parameter. When the nonnegative potential $V\in C (\mathbb{R}^N)$ achieves $0$ with a homogeneous behaviour or on the closure of an open set but remains bounded away from $0$ at infinity, we show the existence of groundstate solutions for small $\varepsilon>0$ and exhibit the concentration behaviour as $\varepsilon\to 0$.Comment: 22 page

Existence, multiplicity and concentration for a class of fractional p&qp\&q Laplacian problems in RN\mathbb{R}^{N}

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1709.0373

Existence of positive solutions for a class of p-Laplacian type generalized quasilinear Schrödinger equations with critical growth and potential vanishing at infinity

In this paper, we study the existence of positive solutions for the following generalized quasilinear Schrödinger equation − div(g p (u)|∇u| p−2∇u) + g p−1 (u)g (u)|∇u| p + V(x)|u| p−2u = K(x)f(u) + Q(x)g(u)|G(u)| p ∗−2G(u), x ∈ R N, where N ≥ 3, 1 < p ≤ N, p Np N−p , g ∈ C1 (R, R+), V(x) and K(x) are positive continuous functions and G(u) = R u 0 g(t)dt. By using a change of variable, we obtain the existence of positive solutions for this problem by using the Mountain Pass Theorem. Our results generalize some existing results