Concave-convex Nonlinearities Research Articles

Multiple solutions for a Kirchhoff-type problem involving nonlocal fractional $p$-Laplacian and concave-convex nonlinearities

This paper examines a class of Kirchhoff nonlocal operators involving concave-convex nonlinearities and sign-changing weight functions. With the aid of the Nehari manifold, the existence of multiple nontrivial nonnegative solutions is obtained.

Quasilinear elliptic problems with concave–convex nonlinearities

In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the [Formula: see text]-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the [Formula: see text]-Laplacian operator is not homogeneous and the nonlinear term is indefinite.

Positive solutions for a Schrödinger–Poisson system involving concave–convex nonlinearities

In this paper, we study the critical growth Schrödinger–Poisson system with a concave term, and establish the existence of multiple positive solutions via using the variational method.

Multiple solutions for the fractional differential equation with concave-convex nonlinearities and sign-changing weight functions

In this paper, by using the fibering map and the Nehari manifold, we prove the existence and multiple results of solutions for the following fractional differential equation: \t\t\t{DTαt(0Dtαu)=λh(t)|u|p−2u+b(t)|u|q−2u,t∈[0,T],u(0)=u(T)=0,\\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} $$\\begin{aligned} \\textstyle\\begin{cases} {_{t}}D{_{T}^{\\alpha}}({_{0}}D{_{t}^{\\alpha}}u)=\\lambda h(t) \\vert u \\vert ^{p-2}u+b(t) \\vert u \\vert ^{q-2}u,\\quad t\\in[0,T],\\\\ u(0)=u(T)=0, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where alphain(frac{1}{2},1), 0< p<2, q>2, lambda>0 and h(t),b(t) are sign-changing continuous functions.

Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

In this paper, we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $\mathcal{L}_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{align*} \mathcal{L}_{K} u + \mu |u|^{q-1}u + \lambda |u|^{p-1}u &= 0 \quad\mbox{in}\quad \Omega, \\ u&=0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{align*} where $\Omega$ is a smooth bounded domain in $ \mathbb R^N $, $N > 2s$, $s\in(0, 1)$, $0 < q < 1 < p\leq \frac{N+2s}{N-2s}$. Moreover, when $\mathcal{L}_K$ reduces to the fractional laplacian operator $-(-\Delta)^s $, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2} (\frac{N+2s}{N-2s}) < q < 1$, $N > 6s$, $ \lambda =1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign changing solution.

Existence of multiple solutions for fractional p-Kirchhoff equations with concave-convex nonlinearities

In this paper, we investigate the existence of multiple solutions for Kirchhoff-type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions as follows: {M(∫R2n|u(x)−u(y)|p|x−y|n+spdxdy)(−△)psu=λ|u|q−2u+αα+β|u|α−2u|v|β,in Ω,M(∫R2n|v(x)−v(y)|p|x−y|n+spdxdy)(−△)psv=μ|v|q−2v+βα+β|v|β−2v|u|α,in Ω,u=v=0,in 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}$$ \\textstyle\\begin{cases} M(\\int_{\\mathbb{R}^{2n}}\\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\\,dx\\, dy)(-\\triangle )_{p}^{s}u=\\lambda|u|^{q-2}u+\\frac{\\alpha}{\\alpha+\\beta}|u|^{\\alpha -2}u|v|^{\\beta}, & \\mbox{in }\\Omega, \\\\ M(\\int_{\\mathbb{R}^{2n}}\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\\,dx\\, dy)(-\\triangle )_{p}^{s}v=\\mu|v|^{q-2}v+\\frac{\\beta}{\\alpha+\\beta}|v|^{\\beta -2}v|u|^{\\alpha}, & \\mbox{in }\\Omega, \\\\ u=v=0, & \\mbox{in }\\mathbb{R}^{n}\\setminus\\Omega, \\end{cases} $$\\end{document} where Ω is a smooth bounded set in mathbb{R}^{n}, n>ps with sin(0,1) fixed, {lambda,mu}>0 are two parameters, 1< q< p< p(tau+1)<alpha+beta<p^{*}, p^{*}=frac{np}{n-sp}, M is a continuous function, given by M(h)=k+lh^{tau}, k>0, l,taugeq0, and (-triangle)_{p}^{s} is the fractional p-Laplacian operator. We will prove that the problem has at least two solutions by using the Nehari manifold method and fibering maps.

具有凸-凹-凸非线性项的边值问题正解的确切个数

本文研究了具有凸-凹-凸非线性项的狄利克雷边值问题正解的分支曲线。通过时间映射分析法，证明了在非线性项为渐近次线性时，边值问题的正解分支曲线为S-型曲线，从而确定了边值问题的正解的确切个数。 In this paper we study the bifurcation curve of the Dirichlet boundary value problem with convex- concave-convex nonlinearities. By using Time-map techniques, we prove that the bifurcation curve of the boundary value problem is S-shaped when the nonlinearities are asymptotic sublinear. Consequently, the exact multiplicity of positive solutions is determined.

Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities

In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, 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.

The (p,q)-elliptic systems with concave-convex nonlinearities

The (p,q)-elliptic systems with concave-convex nonlinearities

Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent

In this paper, we study the existence of multiple positive solutions of the following Schrodinger-Poisson system with critical exponent \begin{document}$\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u, \text{in} \mathbb{R}^{3}, \\-Δφ=l(x)u^{2}, \text{in} \mathbb{R}^{3},\end{cases}\end{equation*}$ \end{document} where \begin{document}$1 and \begin{document}$λ>0 $\end{document} . Under some appropriate conditions on \begin{document}$ l$\end{document} and \begin{document}$h $\end{document} , we show that there exists \begin{document}$λ^{*}>0 $\end{document} such that the above problem has at least two positive solutions for each \begin{document}$λ∈(0,λ^{*}) $\end{document} by using the Mountain Pass Theorem and Ekeland's Variational Principle.

Mixed elliptic problems involving the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$p-$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;Laplacian with nonhomogeneous boundary conditions

In this paper, mixed elliptic problems involving the p-Laplacian and with nonhomogeneous boundary conditions are investigated. At first, the existence of one non-trivial solution, under a suitable behaviour on the nonlinearity and without requiring neither conditions at zero nor conditions at infinity, is established. Then, by adding a condition at infinity on the nonlinearity, also a second non-trivial solution is guaranteed. Some special cases are pointed out as, in particular, the existence of one non-trivial solution when the datum is \begin{document} $(p-1)-$ \end{document} sublinear at zero and the existence of two non-trivial solutions when the nonlinear term is again \begin{document} $(p-1)-$ \end{document} sublinear at zero and, in addition, more than \begin{document} $(p-1)-$ \end{document} superlinear at infinity. As a consequence, the existence of two non-trivial solutions for concave-convex nonlinearities is emphasized. Finally, the case of a simple \begin{document} $(p-1)-$ \end{document} superlinearity at infinity is considered and it is also observed that the same results hold when the nonlinear behaviour, described before for the datum, is instead assumed by the nonhomogeneous Neumann boundary conditions. Concrete examples of applications are also given. The approach is based on variational methods and critical point theory. Precisely, a non-zero local minimum theorem and a two non-zero critical points theorem are applied.

Multiplicity of Solutions for Schrödinger Equations with Concave-Convex Nonlinearities

We study the multiplicity of solutions for a class of semilinear Schrödinger equations: -Δu+V(x)u=gx,u, for x∈RN; u(x)→0, as u→∞, where V satisfies some kind of coercive condition and g involves concave-convex nonlinearities with indefinite signs. Our theorems contain some new nonlinearities.

Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:where Ω is a smooth bounded set in ℝn, n &gt; 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ &gt; 0 are two parameters; the exponent n/(n – 2s) ⩽ q &lt; 2; α &gt; 1, β &gt; 1 satisfy is the fractional critical Sobolev exponent.

Concave-convex nonlinearities for some nonlinear fractional equations involving the Bessel operator

We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.

Critical Nonlocal Systems with Concave-Convex Powers

Abstract By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional p-Laplacian system involving critical concave-convex nonlinearities, provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained although there is no general classification for the optimizers of the critical fractional Sobolev embedding.

Positive solutions for parametric $p$-Laplacian equations

We consider parametric equations driven by the $p$ - Laplacian andwith a reaction which has a $p$ - logistic form or is the sum oftwo competing nonlinearities (concave-convex nonlinearities). Welook for positive solutions and how their solution set depends onthe parameter $\lambda>0$. For the $p$ - logistic equation, weexamine the subdiffusive, equidiffusive and superdiffusive cases.For the equations with competing nonlinearities, we consider thecase of the sum of a ``concave'' (i.e., $(p-1)$ - sublinear) termand of a ``convex'' (i.e., $(p-1)$ - superlinear) term. For thelatter, we do not assume the usual in such casesAmbrosetti-Rabinowitz condition. Our approach is variational basedon the critical point theory combined with suitable truncation andcomparison techniques.

Multiplicity of Positive Solutions for Critical Fractional Equation Involving Concave–Convex Nonlinearities and Sign-Changing Weight Functions

This paper is devoted to study a class of fractional equations with critical exponent, concave nonlinearity and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.

Combined Effects of Concave-Convex Nonlinearities in a Fourth-Order Problem with Variable Exponent

Abstract We study two classes of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving a fourth-order differential operator with variable exponent and power-type nonlinearities. The first result of this paper establishes the existence of a nontrivial weak solution in the case of a small perturbation of the right-hand side. The proof combines variational methods, including the Ekeland variational principle and the mountain pass theorem of Ambrosetti and Rabinowitz. Next we consider a very related eigenvalue problem and we prove the existence of nontrivial weak solutions for large values of the parameter. The direct method of the calculus of variations, estimates of the levels of the associated energy functional and basic properties of the Lebesgue and Sobolev spaces with variable exponent have an important role in our arguments.

Multiplicity of solutions of some quasilinear equations in ${\mathbb{R}^{N}}$ with variable exponents and concave-convex nonlinearities

We prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.

Positive solutions for nonlinear nonhomogeneous Dirichlet problems with concave-convex nonlinearities

We consider a nonlinear parametric Dirichlet equation driven by a nonhomogeneous differential operator involving a reaction exhibiting the competing effects of concave and convex terms. Using variational methods combined with truncation and comparison techniques we prove a bifurcation near zero theorem describing the dependence of the positive solutions on the parameter \(\lambda >0\).