Concave-convex Nonlinearities Research Articles

Multiple nontrivial solutions for a double phase system with concave-convex nonlinearities in subcritical and critical cases

Multiple nontrivial solutions for a double phase system with concave-convex nonlinearities in subcritical and critical cases

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities

Abstract We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by S u ≔ − div ϕ u 2 + ∣ ∇ u ∣ 2 2 ∇ u + ϕ u 2 + ∣ ∇ u ∣ 2 2 u , Su:= -\hspace{0.1em}\text{div}\hspace{0.1em}\left\{\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)\nabla u\right\}+\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)u, where ϕ : [ 0 , + ∞ ) → R \phi :\left[0,+\infty )\to {\mathbb{R}} is a positive continuous function. This operator is introduced by Stuart [Two positive solutions of a quasilinear elliptic Dirichlet problem, Milan J. Math. 79 (2011), 327–341] and depends on not only ∇ u \nabla u but also u u . This particular quasilinear term generally appears in the study of nonlinear optics model, which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution, which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S S : (i) the almost everywhere convergence of the gradient for the minimizing sequence { u n } \left\{{u}_{n}\right\} ; (ii) seeking the reasonable restrictions about S S .

Existence results for the 1-Laplacian problem with a critical concave–convex nonlinearity

Existence results for the 1-Laplacian problem with a critical concave–convex nonlinearity

A comparison of two nonlinear data assimilation methods

Abstract. Advanced numerical data assimilation (DA) methods, such as the four-dimensional variational (4DVAR) method, are elaborate and computationally expensive. Simpler methods exist that take time variability into account, providing the potential of accurate results with a reduced computational cost. Recently, two of these DA methods were proposed for a nonlinear ocean model, an implementation which is costly in time and expertise, developing the need to first evaluate a simpler comparison between these two nonlinear methods. The first method is diffusive back-and-forth nudging (D-BFN), which has previously been implemented in several complex models, most specifically an ocean model. The second is the concave–convex nonlinearity (CCN) method that has a straightforward implementation and promising results with a toy model. D-BFN is less costly than a traditional variational DA system but requires an iterative implementation of equations that integrate the nonlinear model forward and backward in time, whereas CCN only requires integration of the nonlinear model forward in time. This paper investigates if the CCN algorithm can provide competitive results with the already tested D-BFN within simple chaotic models. Results show that observation density and/or frequency, as well as the length of the experiment window, significantly impacts the results for CCN, whereas D-BFN is fairly robust to sparser observations, predominately in time.

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.

Multiple Solutions to the Fractional p-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the L∞-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the L∞-norm. To derive this result, we employ the modified functional method and the dual fountain theorem.

Sign-changing Solutions for Fourth Order Elliptic Equation with Concave-convex Nonlinearities

In this paper, we study the following fourth order elliptic equation: Δ²u - Δu + V(x)u = κ(x)|u|q-2u + |u|p-2u in RN, where Δ² := Δ(Δ) is the biharmonic operator, 4 > N, 1 < q < 2 < p < 2* := 2N / (N - 4). Assuming that V(x) satisfies a class of coercive conditions and the nonnegative weighted function κ(x) belongs to Lp / (p-q)(RN), we obtain the existence of one sign-changing solution with the help of constraint variational method and quantitative deformation lemma. The novelty of this paper is that when the nonlinearity is the combination of concave and convex functions, we are able to obtain the existence of sign-changing solutions. Some recent results are improved and generalized significantly.

A Fractional Elliptic System With Strongly Coupled Critical Terms and Concave-Convex Nonlinearities

By the Nehari method and variational method, two positive solutions are obtained for a fractional elliptic system with strongly coupled critical terms and concave-convex nonlinearities. Recent results from the literature are extended to the fractional case.

Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities

Abstract We investigate the following fractional p-Laplacian convex-concave problem: ( P λ ) ( − Δ ) p s u = λ | u | q − 2 u + | u | p s * − 2 u in Ω , u = 0 in R n \ Ω , $$\left({P}_{\lambda }\right) \begin{cases}\begin{aligned}\hfill {\left(-{\Delta}\right)}_{p}^{s}u& =\lambda \vert u{\vert }^{q-2}u+\vert u{\vert }^{{p}_{s}^{{\ast}}-2}u\hfill & \hfill & \quad \text{in} {\Omega},\hfill \\ \hfill u& =0 \hfill & \hfill & \quad \text{in} {\mathbb{R}}^{n}{\backslash}{\Omega},\hfill \end{aligned}\quad \hfill \end{cases}$$ where Ω is a bounded C 1,1 domain in R n ${\mathbb{R}}^{n}$ , s ∈ (0, 1), p > q > 1, n > sp, λ > 0, and p s * = n p n − s p ${p}_{s}^{{\ast}}=\frac{np}{n-sp}$ is the critical Sobolev exponent. Our analysis extends classical works (A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” J. Funct. Anal., vol. 122, no. 2, pp. 519–543, 1994, B. Barrios, E. Colorado, R. Servadei, and F. Soria, “A critical fractional equation with concave-convex power nonlinearities,” Ann. Inst. Henri Poincare Anal. Non Lineaire, vol. 32, no. 4, pp. 875–900, 2015, J. García Azorero, J. Manfredi, and I. Peral Alonso, “Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations,” Commun. Contemp. Math., vol. 2, no. 3, pp. 385–404, 2000) to fractional p-Laplacian. Owing to the nonlinear and nonlocal properties of ( − Δ ) p s ${\left(-{\Delta}\right)}_{p}^{s}$ , we need to overcome many difficulties and apply notably different approaches, due to the lack of Picone identity, the stability theory, and the strong comparison principle. We show first a dichotomy result: a positive W 0 s , p ( Ω ) ${W}_{0}^{s,p}\left({\Omega}\right)$ solution of (P λ ) exists if and only if λ ∈ (0, Λ] with an extremal value Λ ∈ (0, ∞). The W 0 s , p ( Ω ) ${W}_{0}^{s,p}\left({\Omega}\right)$ regularity for the extremal solution seems to be unknown regardless of whether s = 1 or s ∈ (0, 1). When p ≥ 2, p − 1 < q < p and n > s p ( q + 1 ) q + 1 − p $n{ >}\frac{sp\left(q+1\right)}{q+1-p}$ , we get two positive solutions for (P λ ) with small λ > 0. Here the mountain pass structure is more involved than the classical situations due to the lack of explicit minimizers for the Sobolev embedding, we should proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level. Finally, we show another new result for (P λ ) and all p > q > 1: without sign constraint, (P λ ) possesses infinitely many solutions when λ > 0 is small enough. Here we use the Z 2 ${\mathbb{Z}}_{2}$ -genus theory, based on a space decomposition for reflexible and separable Banach spaces, which has its own interest.

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