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In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with (p, q)-Laplacian:... [ABSTRACT FROM AUTHOR]

In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian: $ \begin{equation*} \begin{cases} -\Delta_{p}u-\Delta_{q}u+\phi|u|^{q-2} u = f\left(x, u\right)+\mu|u|^{s-2} u & \text { in } \Omega, \\ -\Delta \phi = |u|^q & \text { in } \Omega, \\ u = \phi = 0 & \text { on } \partial \Omega, \end{cases} \end{equation*} $ where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, $ \mu > 0, N > 1 $, and $ -\Delta_{{\wp}}\varphi = div(|\nabla\varphi|^{{\wp}-2} \nabla\varphi) $, with $ {\wp}\in \{p, q\} $, is the homogeneous $ {\wp} $-Laplacian. $ 1 < p < q < \frac{q^*}{2} $, $ q^*: = \frac{Nq}{N-q} < s $, and $ q^* $ is the critical exponent to $ q $. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $ (p, q) $-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.

In this paper, we consider a critical (p, q) equation on the Heisenberg group of the following form: - Δ H , p u - Δ H , q u + V (ξ) (| u | p - 2 u + | u | q - 2 u) = μ ∫ H n F (ξ , u) | η - 1 ξ | λ d ξ f (η , u) + | u | q ∗ - 2 u , where the operator - Δ H , ℘ φ = div H (| D H φ | H ℘ - 2 D H φ) , with ℘ ∈ { p , q } , is the proverbial horizontal ℘ -Laplacian on the Heisenberg group, 1 < p < (2 Q - λ) 2 Q q < q < Q , q ∗ = q Q / (Q - q) is the critical exponent, and Q = 2 n + 2 is the homogeneous dimension of H n , μ and λ are some real parameters. Under the appropriate assumptions of potential functions V and f, the existence of entire solutions to the above equation on the Heisenberg group is obtained by using the mountain pass theorem and the concentration compactness principle. The results presented here extend or complete recent papers and are new to critical equations involving (p, q)-Laplacian operators and convolution terms on Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of a critical (p, q) equation with Hardy terms on the Heisenberg group. Existence of entire solutions is obtained via an application of some concentration–compactness type results and the mountain pass theorem. Our results are presented in the model case of the (p, q) horizontal Laplacian equations, but the method can be extended to deal with a more general class of problems with operators of (p, q) growth. [ABSTRACT FROM AUTHOR]

We deal with sign-changing solutions for the Kirchhoff equation $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ u = 0, \; \ x\in \partial\Omega, \end{cases} $ where $ a, b > 0 $ and $ \lambda, \mu\in\mathbb{R} $ being parameters, $ \Omega\subset \mathbb{R}^{3} $ is a bounded domain with smooth boundary $ \partial\Omega $. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists $ \mu^{\ast} > 0 $ such that above problem has at least a least energy sign-changing (or nodal) solution if $ \lambda < a\lambda_{1} $ and $ \mu > \mu^{\ast} $, where $ \lambda_{1} > 0 $ is the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. It is noticed that the nonlinearity $ \lambda u+\mu|u|^{2}u $ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.

In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity \begin{equation*} \left\{ \begin{array} {ll} \left(a+b \iint_{\Omega^2} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dxdy\right)(-\Delta)^s u + V(x)u = Q(x) |u|^{p-2}u \ln u^2, & {\rm in } \ \Omega, \\ u=0, & {\rm in } \ \mathbb{R}^N \setminus \Omega, \end{array} \right. \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 2s$ ($0 < s < 1$), $(-\Delta)^s$ is the fractional Laplacian, $V, Q$ are continuous, $V, Q \ge 0$. $a, b > 0$ are constants, $4 < p < 2_s^* := \frac{2N}{N-2s}$. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.

In this paper, we consider a Neumann problem of Kirchhoff type equation \begin{equation*} \begin{cases} \displaystyle-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u= Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle\frac{\partial u}{\partial v}=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^3$ is a bounded domain with a smooth boundary, $a,b>0$, $10$ is a real parameter, $Q(x)$ and $P(x)$ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions.

This paper deals with following Kirchhoff-type system with critical growth \[\begin{cases} -(a+ b\int _{\mathbb{R}^3}|\nabla u|^{2}dx)\Delta u+ V(x)u+\phi|u|^{p-2}u =|u|^{4}u+\mu f(u), ~\ x\in\mathbb{R}^3,\\ (-\Delta)^{\alpha/2}\phi=l|u|^p, ~\ x\in \mathbb{R}^3, \end{cases}\] where $a, \mu>0$, $b, l\geq0$, $\alpha\in(0,3)$, $p\in[2,3)$ and $\phi|u|^{p-2}u$ is a Hartree-type nonlinearity. By the minimization argument on the nodal Nehari manifold and the quantitative deformation lemma, we prove that the above system has a least energy nodal solution. Our result improve and generalize some interesting results which were obtained in subcritical case.

"In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity\begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u=|u|^{q-2}u\ln u^2, ~x\in\Omega \\ u=0, ~\ x\in \partial\Omega, \end{array} \right.\end{equation*}where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b>0$ are constant, 4 ≤ 2p < q < p* and N > p. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains."

Abstract In this paper, we study the following Kirchhoff–Schrödinger–Poisson systems: {−(a+b∫R3|∇u|2dx)Δu+V(x)u+ϕu=f(u),x∈R3,−Δϕ=u2,x∈R3, $$\textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=f(u), &x \in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}, &x\in \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, V∈C(R3,R+) $V\in \mathcal{C}(\mathbb{R} ^{3},\mathbb{R}^{+})$. By using constraint variational method and the quantitative deformation lemma, we obtain a least-energy sign-changing (or nodal) solution ub $u_{b}$ to this problem, and study the energy property of ub $u_{b}$. Moreover, we investigate the asymptotic behavior of ub $u_{b}$ as the parameter b↘0 ${b\searrow 0}$.

In this paper, we study the following fourth-order Kirchhoff-type equation Δ 2 u - a + b ∫ R N | ∇ u | 2 d x Δ u + V (x) u = K (x) f (u) , x in R N , with the potential V(x) vanishing at infinity. Under suitable conditions, by using constraint variational method and the quantitative deformation lemma, we obtain a least energy sign-changing (or nodal) solution to this problem. Moreover, we prove that this least energy sign-changing solution has precisely two nodal domains. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger-Poisson system { − Δ u + V (x) u + λ ϕ u = | u | 4 u + μ f (u) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where V (x) is a smooth function and μ , λ > 0. Under suitable conditions on f , by using constraint variational method and the quantitative deformation lemma, if μ is large enough, we obtain a least-energy sign-changing (or nodal) solution u λ to this problem for each λ > 0 , and its energy is strictly larger than twice that of the ground state solutions. Moreover, we study the asymptotic behavior of u λ as the parameter λ ↘ 0. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equation \begin{equation*} \begin{cases} - \Delta u +V(x)u = -\lambda Q(x)|u|^{q-2}u+ Q(x)f(u),\quad x\in\mathbb{R}^N,\\ u(x)\rightarrow 0,\quad \hbox{as}\ |x|\rightarrow +\infty,\end{cases} \tag*{(P)$_\lambda$} \end{equation*} where $1

In the present paper the general variation principle of plane stress state of micropolar theory of elasticity is considered in a circular area, on the basis of which the basic equations and boundary conditions of the mentioned theory are obtained. Accepting the known hypotheses of the construction of the theory of micropolar elastic thin straight bars, plates and shells, general variation principle for applied model of micropolar elastic circular thin bars with transverse shear deformations is obtained on the basis of variation principle of plane stress state. Based on the constructed variation principle the basic equations and natural boundary conditions of applied model of micropolar elastic circular thin bar are obtained. It is confirmed that all energy theorems and Ritz, Bubnov-Galerkin, FEM variation methods are applicable for the constructed model of micropolar elastic circular thin bar and for solutions of corresponding boundary value problems of the applied model. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equation ... where ... and Q are radial positive functions, which can be vanishing or coercive at infinity, f is asymptotically linear or superlinear at infinity. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of nontrivial standing wave solution of the discrete nonlinear Schrödinger equation with the growing potential at infinity. Firstly we derive a discrete version of compact embedding theorem. Then combining the Nehari manifold approach and the compact embedding theorem we show the existence of nontrivial standing wave solution without Palais-Smale condition. We also prove the exponential decay of the standing wave solutions. [ABSTRACT FROM AUTHOR]