Constant sign and nodal solutions for superlinear double phase problems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Constant Sign ◽  
Nodal Solution ◽  
Negative Solution ◽  
Nontrivial Solutions ◽  
Nodal Solutions ◽  
Nodal Domains ◽  
Double Phase ◽  
Variational Tools ◽  
Nehari Method ◽  
Superlinear Reaction

Abstract We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

Multiple solutions for parametric double phase Dirichlet problems

2020 ◽  
pp. 2050006 ◽  
Author(s):  
N. S. Papageorgiou ◽  
C. Vetro ◽  
F. Vetro
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Parameter ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools

We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter [Formula: see text] precisely in terms of the spectrum of the [Formula: see text]-Laplacian.

Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

2018 ◽  
Vol 18 (1) ◽  
pp. 17-40
Author(s):  
Yinbin Deng ◽  
Shuangjie Peng ◽  
Jixiu Wang
Keyword(s):  
Elliptic Equations ◽  
Order Term ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Content Type ◽  
Nodal Domains ◽  
Change Of Variables ◽  
Graphic Content ◽  
Quasilinear Elliptic

AbstractThis paper is concerned with the following type of quasilinear elliptic equations in{\mathbb{R}^{N}}involving thep-Laplacian and critical growth:-\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u,which arises as a model in mathematical physics, where{2<p<N},{p^{*}=\frac{Np}{N-p}}. For any given integer{k\geq 0}, by using change of variables and minimization arguments, we obtain, under some additional assumptions onpandq, a radial sign-changing nodal solution with{k+1}nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

scholarly journals Nonlinear Dirichlet problems with unilateral growth on the reaction

2019 ◽  
Vol 31 (2) ◽  
pp. 319-340
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Dirichlet Problem ◽  
Critical Groups ◽  
Constant Sign ◽  
Nodal Solution ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Multiplicity Results ◽  
Subcritical Growth ◽  
Nonlinear Dirichlet Problem ◽  
The Third

AbstractWe consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, {p=2}), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.

scholarly journals CONSTANT SIGN AND NODAL SOLUTIONS FOR NONLINEAR ROBIN EQUATIONS WITH LOCALLY DEFINED SOURCE TERM

2020 ◽  
Vol 25 (3) ◽  
pp. 374-390
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Differential Operator ◽  
Source Term ◽  
Constant Sign ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Nodal Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Special Cases ◽  
Variational Tools ◽  
Nonlinear Nonhomogeneous Differential Operator

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).

scholarly journals Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jianqing Chen ◽  
Qian Zhang
Keyword(s):  
Soliton Solutions ◽  
Nodal Solution ◽  
Entire Space ◽  
Nodal Solutions ◽  
Moser Iteration ◽  
Nodal Domains ◽  
Alpha Beta ◽  
Beta 2 ◽  
Principle Of Symmetric Criticality ◽  
Schrödinger System

<p style='text-indent:20px;'>This paper is concerned with the following quasilinear Schrödinger system in the entire space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{N}(N\geq3) $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &amp;-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ &amp; u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \alpha,\beta&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 2&lt;\alpha+\beta&lt;2^* = \frac{2N}{N-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ k &gt;0 $\end{document}</tex-math></inline-formula> is a parameter. By using the principle of symmetric criticality and the moser iteration, for any given integer <inline-formula><tex-math id="M5">\begin{document}$ \xi\geq2 $\end{document}</tex-math></inline-formula>, we construct a non-radially symmetrical nodal solution with its <inline-formula><tex-math id="M6">\begin{document}$ 2\xi $\end{document}</tex-math></inline-formula> nodal domains. Our results can be looked on as a generalization to results by Alves, Wang and Shen (Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259 (2015) 318-343).</p>

scholarly journals Multiple nodal solutions for the Schr\”odinger-Poisson system with an asymptotically cubic term

2021 ◽  
Author(s):  
Hui Guo ◽  
Ronghua Tang ◽  
Tao Wang
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Nonlinear Term ◽  
Affirmative Answer ◽  
Poisson System ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Nodal Domains ◽  
Deformation Lemma ◽  
Asymptotically Cubic

This paper deals with the following Schr\“odinger-Poisson system \begin{equation}\left\{\begin{aligned} &-\Delta u+u+ \lambda\phi u=f(u)\quad\mbox{in }\mathbb{R}^3,\\ &-\Delta \phi=u^{2}\quad\mbox{in }\mathbb{R}^3, \end{aligned}\right.\end{equation} where $\lambda>0$ and $f(u)$ is a nonlinear term asymptotically cubic at the infinity. Taking advantage of the Miranda theorem and deformation lemma, we combine some new analytic techniques to prove that for each positive integer $k,$ system \eqref{zhaiyaofc} admits a radial nodal solution $U_k^{\lambda}$, which has exactly $k+1$ nodal domains and the corresponding energy is strictly increasing in $k$. Moreover, for any sequence $\{\lambda_n\}\to 0_+$ as $n\to\infty,$ up to a subsequence, $U_k^{\lambda_n}$ converges to some $U_k^0\in H_r^1(\mathbb{R}^3)$, which is a radial nodal solution with exactly $k+1$ nodal domains of \eqref{zhaiyaofc} for $\lambda=0 $. These results give an affirmative answer to the open problem proposed in [Kim S, Seok J. Commun. Contemp. Math., 2012] for the Schr\”odinger-Poisson system with an asymptotically cubic term.

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

scholarly journals Anisotropic singular double phase Dirichlet problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rǎdulescu ◽  
Youpei Zhang
Keyword(s):  
Positive Solutions ◽  
Singular Term ◽  
Independent Interest ◽  
Phase Problem ◽  
Type Result ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools ◽  
Superlinear Perturbation ◽  
Double Phase Problem

<p style='text-indent:20px;'>We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on <inline-formula><tex-math id="M1">\begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}</tex-math></inline-formula>. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.</p>

scholarly journals Constant sign and nodal solutions for parametric anisotropic (p, 2) -equations

2021 ◽  
pp. 1-18
Author(s):  
Nikolaos S. Papageorgiou ◽  
Dušan D. Repovš ◽  
Calogero Vetro
Keyword(s):  
Constant Sign ◽  
Nodal Solutions
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