Concave-convex Nonlinearities Research Articles

Nehari manifold approach for fractional p(.)-Laplacian system involving concave-convex nonlinearities

In this article, using Nehari manifold method we study the multiplicity of solutions of the nonlocal elliptic system involving variable exponents and concave-convex nonlinearities, $$\displaylines{ (-\Delta)_{p(\cdot)}^{s} u=\lambda a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \quad x\in \Omega; \cr (-\Delta)_{p(\cdot)}^{s} v=\mu b(x)| v|^{q(x)-2}v+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| v|^{\alpha(x)-2}v| u| ^{\beta(x)},\quad x\in \Omega; \cr u=v=0,\quad x\in \Omega^c:=\mathbb R^N\setminus\Omega, }$$ where \(\Omega\subset\mathbb R^N\), \(N\geq2\) is a smooth bounded domain, \(\lambda,\mu>0\) are parameters, and \(s\in(0,1)\). We show that there exists \(\Lambda>0\) such that for all \(\lambda+\mu<\Lambda\), this system admits at least two non-trivial and non-negative solutions under some assumptions on \(q,\alpha,\beta,a,b,c\).
 For more information see https://ejde.math.txstate.edu/Volumes/2020/98/abstr.html

On Schrödinger–Poisson systems involving concave–convex nonlinearities via a novel constraint approach

In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: [Formula: see text] where [Formula: see text] are two parameters, [Formula: see text], [Formula: see text] is a potential well, [Formula: see text] and [Formula: see text]. Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on [Formula: see text] due to [Formula: see text]. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.

The Nehari manifold for fractional p-Laplacian system involving concave–convex nonlinearities and sign-changing weight functions

ABSTRACT In this paper, we consider a fractional p-Laplacian system with both concave–convex nonlinearities and sign-changing weight functions in bounded domains: By using the Nehari manifold, together with Ekeland's variational principle, we prove that the above system has at least two nontrivial solutions when the pair of the parameters belongs to a certain subset of .

Nodal solution for Kirchhoff-type problems with concave-convex nonlinearities

In this paper, we study the existence of nodal solutions for the Kirchhoff-type problem with concave-convex nonlinearities where Ω is a bounded domain with smooth boundary in , a . By constraining the energy functional of the above problem on a subset of the nodal Nehari set , we show that there exists a constant such that for any , the above problem has a nodal solution with positive energy.

Multiple positive solutions for singular elliptic problems involving concave-convex nonlinearities and sign-changing potential

In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, $$0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}$$, 1 < q < 2 < p < 2* and $$2* = \frac{{2N}}{{N - 2}}$$ is the Sobolev critical exponent, the coefficient functions f, g may change sign on Ω. By the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

Multiple positive solutions for biharmonic equation of Kirchhoff type involving concave-convex nonlinearities

In this article, we study the multiplicity of positive solutions for the biharmonic equation of Kirchhoff type involving concave-convex nonlinearities, $$ \Delta^2u-\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)\Delta u+V(x)u =\lambda f_1(x)|u|^{q-2}u+f_2(x)|u|^{p-2}u. $$ Using the Nehari manifold, Ekeland variational principle, and the theory of Lagrange multipliers, we prove that there are at least two positive solutions, one of which is a positive ground state solution. For more information see https://ejde.math.txstate.edu/Volumes/2020/44/abstr.html

Existence and multiplicity of solutions for Nonlocal elliptic problem involving p−biharmonic operator and concave-convex nonlinearities

Existence and multiplicity of solutions for Nonlocal elliptic problem involving p−biharmonic operator and concave-convex nonlinearities

New multiple solutions for a Schrödinger–Poisson system involving concave-convex nonlinearities

In this paper, we study the following critical growth Schrödinger-Poisson system with concave-convex nonlinearities term $\left\{\begin{array} -\Delta u + u + \eta\varphi u = \lambda f(x) u^{q-1} + u^5, in R^3, \\ -\Delta \varphi = u^2, in R^3,\end{array}\right. $ where $1 < q < 2, \eta\in \mathbb{R}, \lambda > 0$ is a real parameter and $f \in L^{\frac{6}{6-q}} (\mathbb{R}^3)$ is a nonzero nonnegative function. Using the variational method, we obtain that there exists a positive constant $\lambda_* > 0$ such that for all $\lambda \in (0,\lambda_*)$, the system has at least two positive solutions.

Multiplicity of Positive Solutions for a Nonlocal Elliptic Problem Involving Critical Sobolev-Hardy Exponents and Concave-Convex Nonlinearities

In this article, we study the following critical problem involving the fractional Laplacian: $$\begin{cases}(-\Delta)^{\frac{\alpha}{2}}u-\gamma\frac{u}{|x|^\alpha}=\lambda\frac{|u|^{q-2}}{|x|^s}+\frac{u^{2_\alpha^*(t)-2}u}{|x|^t} & in\;\Omega, \\u=0 & in\;\mathbb{R}^N\backslash\Omega,\end{cases}$$ where Ω ⊂ ℝN (N > α) is a bounded smooth domain containing the origin, α ∈ (0,2), 0 ≤ s, t 0, $$2_\alpha^*(t)=\frac{2(N-t)}{N-\alpha}$$ is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γH, and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.

Existence and multiplicity of solutions to fractional p-Laplacian systems with concave–convex nonlinearities

This paper is concerned with a fractional [Formula: see text]-Laplacian system with both concave–convex nonlinearities. The existence and multiplicity results of positive solutions are obtained by variational methods and the Nehari manifold.

Multiple solutions for fractional p-Laplace equation with concave-convex nonlinearities

In this paper, we investigate the existence of solutions for the fractional p-Laplace equation (−Δ)psu+V(x)|u|p−2u=h1(x)|u|q−2u+h2(x)|u|r−2uin RN,\\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}$$ (-\\Delta)_{p}^{s}u+V(x) \\vert u \\vert ^{p-2}u=h_{1}(x) \\vert u \\vert ^{q-2}u+h_{2}(x) \\vert u \\vert ^{r-2}u \\quad \\mbox{in } \\mathbb{R}^{N}, $$\\end{document} where N>sp, 0< s<1<p, 1< q< p< r< p_{s}^{*}:=frac{Np}{N-sp}, and the potential function V(x)>0 and h_{1}(x), h_{2}(x) are allowed to change sign in mathbb {R}^{N}. By using variant fountain theorem, we prove that the above equation admits infinitely many small and high energy solutions.

Topological properties of the solution sets for parametric nonlinear Dirichlet problems

ABSTRACT The goal of this paper is to investigate a parametric Dirichlet problem with -Laplacian and concave–convex nonlinearity. Denoting by the set of positive solutions of the problem corresponding to the parameter , we prove that the set is compact in the -topolgy. We also establish an upper semicontinuity and a Mosco convergence result for the multivalued mapping .

Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave\u2013convex nonlinearities

We consider the combined effect of concave–convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When alpha+ beta<4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais–Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.

Multiplicity of Positive Solutions for Kirchhoff Systems

The paper studies the combined effect of concave–convex nonlinearities on the number of solutions for a Kirchhoff system, which involves weight functions and critical exponents. Using Nehari manifold, the existence of multiple positive solutions is established when the pair of the parameters is under a certain range.

Nonlocal Elliptic Systems Involving Critical Sobolev-Hardy Exponents and Concave-convex Nonlinearities

In this paper, a system of fractional elliptic equation is investigated, which involving fractional critical Sobolev-Hardy exponent and concave-convex terms. By means of variational methods and analytic techniques, the existence and multiplicity of positive solutions to the system is established.

Existence and concentration of positive solutions for non-autonomous Schrödinger-Poisson systems

In this paper, we deal with a class of non-autonomous Schrödinger-Poisson systems involving concave-convex nonlinearities and weight functions. We obtain the existence and multiplicity of positive solutions by a global compactness lemma of Schrödinger-Poisson type and Ljusternik-Schnirelmann theory. Moreover, we also conclude that the global maximum point of the solution concentrates on a local minimum point of the weight function.

Multiple solutions for quasilinear elliptic problems with concave-convex nonlinearities in Orlicz\u2013Sobolev spaces

Using variational arguments, we establish the existence of two nontrivial solutions for quasilinear elliptic problems in Orlicz–Sobolev spaces, where the nonlinear terms exhibit the combined effects of concave and convex without the Ambrosetti–Rabinowitz type condition.

Solutions for fourth-order Kirchhoff type elliptic equations involving concave–convex nonlinearities in [formula omitted

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations Δ2u−M(‖∇u‖22)Δu+V(x)u=f(x,u),x∈RN,where M(t):R→R is the Kirchhoff function, f(x,u)=λk(x,u)+h(x,u), λ≥0, k(x,u) is of sublinear growth and h(x,u) satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for λ=0. For λ>0 small enough, we obtain at least two nontrivial solutions. Furthermore, if f(x,u) is odd in u, we show that above equations possess infinitely many solutions for all λ≥0. Our theorems generalize some known results in the literatures even for λ=0 and our proof is based on the variational methods.

Solutions to a gauged Schrödinger equation with concave–convex nonlinearities without (AR) condition

In this paper, we are concerned with a gauged nonlinear Schrödinger equation where are positive parameters and Under appropriate assumptions on the potential V and nonlinear terms g and f, where g has sublinear growth and f has asymptotically linear or superlinear growth without (AR) condition, we acquire the existence and multiplicity of solutions by means of Mountain pass theorem and Fountain theorem. Our results generalize and improve the recent result in the literature.