Concave-convex Nonlinearities Research Articles

Discrete Schrödinger equations and systems with mixed and concave-convex nonlinearities

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models.

Results on the Existence and Multiple Solutions of a p-Kirchhoff Equation with a Subcritical Hardy-Sobolev Exponent and Concave-Convex Nonlinearities

Results on the Existence and Multiple Solutions of a p-Kirchhoff Equation with a Subcritical Hardy-Sobolev Exponent and Concave-Convex Nonlinearities

Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities

In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1−Δpu+V(x)|u|p−2u−Δp(|u|2α)|u|2α−2u=λh1(x)|u|m−2u+h2(x)|u|q−2u,x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned}& -\\Delta _{p}u+V(x) \\vert u \\vert ^{p-2}u-\\Delta _{p}\\bigl( \\vert u \\vert ^{2\\alpha}\\bigr) \\vert u \\vert ^{2\\alpha -2}u= \\lambda h_{1}(x) \\vert u \\vert ^{m-2}u+h_{2}(x) \\vert u \\vert ^{q-2}u, \\\\& \\quad x\\in {\\mathbb{R}}^{N}, \\end{aligned}$$ \\end{document} where Δpu=div(|∇u|p−2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Delta _{p}u=\\operatorname{div}(|\ abla u|^{p-2}\ abla u)$\\end{document}, 1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< p< N$\\end{document}, λ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\ge 0$\\end{document}, and 1<m<p<2αp<q<2αp∗=2αpNN−p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1< m< p<2\\alpha p<q<2\\alpha p^{*}=\\frac{2\\alpha pN}{N-p}$\\end{document}. The functions V(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$V(x)$\\end{document}, h1(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{1}(x)$\\end{document}, and h2(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{2}(x)$\\end{document} satisfy some suitable conditions. Using variational methods and some special techniques, we prove that there exists λ0>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{0}>0$\\end{document} such that Eq. (0.1) admits infinitely many high energy solutions in W1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W^{1,p}({\\mathbb{R}}^{N})$\\end{document} provided that λ∈[0,λ0]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in [0,\\lambda _{0}]$\\end{document}.

The Multiplicity of Nonnegative Nontrivial Solutions for p(x)-Kirchhoff Equation with Concave–Convex Nonlinearities

The Multiplicity of Nonnegative Nontrivial Solutions for p(x)-Kirchhoff Equation with Concave–Convex Nonlinearities

Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem as the main tool. The second aim is to obtain that our problem admits a sequence of infinitely many small-energy solutions. To obtain these results, we utilize the dual fountain theorem. In addition, we prove the existence of a sequence of infinitely many weak solutions converging to 0 in L∞-space. To derive this result, we exploit the dual fountain theorem and the modified functional method.

Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

Fractional Musielak spaces: a class of non-local problem involving concave–convex nonlinearity

Fractional Musielak spaces: a class of non-local problem involving concave–convex nonlinearity

On a critical elliptic system with concave–convex nonlinearities

On a critical elliptic system with concave–convex nonlinearities

Solutions of a Schrödinger–Kirchhoff–Poisson system with concave–convex nonlinearities

Solutions of a Schrödinger–Kirchhoff–Poisson system with concave–convex nonlinearities

Sign-changing solutions for a class of Schrödinger-Bopp-Podolsky system with concave-convex nonlinearities

In this article we are devoted to deal with the following Schrödinger-Bopp-Podolsky system{−Δu+V(x)u+ϕu=f(x)|u|q−2u+|u|p−2u,x∈R3,−Δϕ+a2Δ2u=4πu2,x∈R3, where 1<q<2, 4<p<6 and a≥0. Assuming that V(x) satisfies the typically coercive condition and the nonnegative potential f(x) belongs to L66−q(R3) with an appropriate upper bound, we show the existence of sign-changing solutions with positive energy, by means of constraint variational method and quantitative deformation lemma.

Infinitely many positive energy solutions for an elliptic equation involving critical Sobolev growth, Hardy potential and concave–convex nonlinearity

Infinitely many positive energy solutions for an elliptic equation involving critical Sobolev growth, Hardy potential and concave–convex nonlinearity

Existence of non-negative solutions for a system with concave-convex nonlinearity

Existence of non-negative solutions for a system with concave-convex nonlinearity

Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent

Aim of this paper is to discuss the existence of multiple solutions to double phase anisotropic variational problems for the case of a combined effect of concave-convex nonlinearities. Especially the superlinear (convex) term to the given problem substantially fulfills a weaker condition as well as Ambrosetti-Rabinowitz condition. To achieve these results, we apply the variational methods such as the famous mountain pass theorem and Ekeland's type variational principle when an energy functional corresponding to our problem satisfies the compactness condition of the Palais-Smale type. In particular, we establish several existence results of a sequence of infinitely many solutions by employing the Cerami compactness condition. The key tools for obtaining these results are the fountain theorem and the dual fountain theorem.

Nonnegative nontrivial solutions for a class of p(x)-Kirchhoff equation involving concave-convex nonlinearities

In this paper, we study the existence of a class of p(x)-Kirchhoff equation involving concave-convex nonlinearities. The main tools used are the perturbation technique, variational method, and a priori estimation.

Singular Anisotropic Problems with Competition Phenomena

We consider a Dirichlet problem driven by the anisotropic (p(z), q(z))-Laplacian, with a parametric reaction exhibiting the combined effects of singular and concave-convex nonlinearities. The superlinear term may change sign. Using variational tools together with truncation and comparison techniques, we prove a global (for the parameter lambda >0) existence and multiplicity theorem (a bifurcation-type theorem).

Infinitely Many Small Energy Solutions to the Double Phase Anisotropic Variational Problems Involving Variable Exponent

This paper is devoted to double phase anisotropic variational problems for the case of a combined effect of concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, on a class of superlinear term which is different from the previous related works, is to discuss the multiplicity result of non-trivial solutions by applying the dual fountain theorem as the main tool. In particular, our main result is obtained without assuming the conditions on the nonlinear term at infinity.

Existence of solutions to a 1-Laplacian problem with a concave-convex nonlinearity

In this paper, we analyze a “concave-convex” type problem involving the 1-Laplacian operator in a general Lipschitz–continuous domain and prove the existence of two positive solutions. Owing to 1-Laplacian is 0-homogeneous, the “concave” term must be singular. Hence, we should deal with an energy functional having two non–differentiable terms: the total variation and that one coming from the singular term. Due to these difficulties, we do not get solutions as critical points of the energy functional defined in the BV(Ω) space. Instead, we study problems involving the p-Laplacian operator and let p go to 1.

Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional r(·)-Laplacian in RN

This paper is concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional r(·)-Laplacian equations of Kirchhoff–Schrödinger type with concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, under suitable assumptions on a nonlinear term, is to discuss the multiplicity result of non-trivial solutions by using the dual fountain theorem as the main tool.

Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Abstract In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem { − Δ u = | u | 2 ∗ ( t ) − 2 u | y | t + μ | u | q − 2 u in Ω , u = 0 on ∂ Ω , \begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u &amp; \text { in } \Omega, \\ u=0 &amp; \text { on } \partial \Omega,\end{cases} where Ω is an open bounded domain in ℝ N , which contains some points (0,z*), μ &gt; 0 , 1 &lt; q &lt; 2 , 2 ∗ ( t ) = 2 ( N − t ) N − 2 \mu&gt;0,1&amp;#x003C;q&amp;#x003C;2,2^*(t)=\frac{2(N-t)}{N-2} , 0 ≤ t &lt; 2, x = (y, z) ∈ ℝ k × ℝ N−k , 2 ≤ k ≤ N. We prove that if N &gt; 2 q + 1 q − 1 + t $N &gt; 2{{q + 1} \over {q - 1}} + t$ , then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.

Infinitely many solutions for an elliptic equation in divergent form with critical Sobolev exponent and concave-convex nonlinearity

Infinitely many solutions for an elliptic equation in divergent form with critical Sobolev exponent and concave-convex nonlinearity