On Bobkov-Tanaka type spectrum for the double-phase operator

A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with [formula omitted]-Laplacian

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta |u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $\Omega$. The main result of our research is the construction of a continuous curve in $(\alpha,\beta)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $\Omega$ for which the corresponding first Dirichlet eigenvalue of $-\Delta_p$ is not monotone w.r.t. $p > 1$.

Nonzero positive weak solutions of systems of [formula omitted]-Laplace equations

In this paper, we investigate singular \(p\)-Laplacian equations of the form \(\Delta _pu + f(x,\nabla u)u^{-\lambda} = 0\) with zero Dirichlet boundary condition in a ball \(B \subset R^N\); where \(p > 1, \lambda > 0\), and give a sufficient condition for the equation to have a positive solution, by means of a supersolution and a subsolution.

In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.

In this paper, we study the following critical system with fractional Laplacian: \t\t\t{(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (-\\Delta)^{s}u+\\lambda_{1}u=\\mu_{1}|u|^{2^{\\ast}-2}u+\\frac{\\alpha \\gamma}{2^{\\ast}}|u|^{\\alpha-2}u|v|^{\\beta} & \\text{in } \\Omega, \\\\ (-\\Delta)^{s}v+\\lambda_{2}v= \\mu_{2}|v|^{2^{\\ast}-2}v+\\frac{\\beta \\gamma}{2^{\\ast}}|u|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega, \\\\ u=v=0 & \\text{in } \\mathbb{R}^{N}\\setminus\\Omega, \\end{cases} $$\\end{document} where (-Delta)^{s} is the fractional Laplacian, 0< s<1, mu_{1},mu_{2}>0, 2^{ast}=frac{2N}{N-2s} is a fractional critical Sobolev exponent, N>2s, 1<alpha, beta<2, alpha+beta=2^{ast}, Ω is an open bounded set of mathbb{R}^{N} with Lipschitz boundary and lambda_{1},lambda_{2}>-lambda_{1,s}(Omega), lambda_{1,s}(Omega) is the first eigenvalue of the non-local operator (-Delta)^{s} with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all gamma>0. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when gammarightarrow0.

A $p$-Laplacian equation with weak Allee effect growth rate and Dirichlet boundary condition is considered. The existence, multiplicity and bifurcation of positive solutions are proved with comparison and variational techniques. The existence of multiple positive solutions implies that the related ecological system may exhibit bistable dynamics.

Nontrivial solutions of discrete nonlinear equations with variable exponent

We study the following class of linearly coupled Schrödinger elliptic systems [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term [Formula: see text] by the assumption [Formula: see text], for some [Formula: see text]. We deal with three cases: Firstly, we study the subcritical case, [Formula: see text], and we prove the existence of positive ground state for all parameter [Formula: see text]. Secondly, we consider the critical case, [Formula: see text], and we prove that there exists [Formula: see text] such that the coupled system possesses positive ground state solution for all [Formula: see text]. In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case [Formula: see text], and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.

The Nehari manifold for fractional [formula omitted]-Laplacian equation with logarithmic nonlinearity on whole space

We study the fractional p-Laplacian problem with Hardy–Sobolev exponents. We prove: there is a λ0>0 such that for any λ∈(0,λ0), the above problem possesses infinitely many solutions. We achieve our goal by making use of variational methods, more specifically, the Nehari manifold and Lusternik–Schnirelmann theory.

Positive solutions are obtained for the boundary value problemHeref (t, u)≥–M, (Mis a positive constant) for (t, u) ∈ [0, 1]×(0, ∞). We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

Existence of multiple solutions for an elliptic system with sign-changing weight functions

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter β is sufficiently large.

In this paper we study optimal control problems with the regional fractional \begin{document} $p$\end{document} -Laplace equation, of order \begin{document} $s \in \left( {0,1} \right)$ \end{document} and \begin{document} $p \in \left[ {2,\infty } \right)$ \end{document} , as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the regional fractional \begin{document} $p$\end{document} -Laplace operator. We show existence and uniqueness of solutions to the state equations and existence of solutions to the optimal control problems. We prove that the regional fractional \begin{document} $p$\end{document} -Laplacian approaches the standard \begin{document} $p$\end{document} -Laplacian as \begin{document} $s$ \end{document} approaches 1. In this sense, this fractional \begin{document} $p$\end{document} -Laplacian can be considered degenerate like the standard \begin{document} $p$\end{document} -Laplacian. To overcome this degeneracy, we introduce a regularization for the regional fractional \begin{document} $p$\end{document} -Laplacian. We show existence and uniqueness of solutions to the regularized state equation and existence of solutions to the regularized optimal control problem. We also prove several auxiliary results for the regularized problem which are of independent interest. We conclude with the convergence of the regularized solutions.