In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $κ\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $λ$ and on $κ$.
Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $λ\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{κ+1}-1$ solutions for sufficiently negative $λ$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $λ$ and the profiles of positive solutions.
Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.
This paper is concerned with multiplicity results for parametric singular double phase problems in $\mathbb{R}^N$ via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.
We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation $$ \begin{array}{c} -Δu +V(x)u=λa(x)u^{-γ}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array} $$ beyond the Nehari extremal value, as defined in Il'yasov [Topol. Methods Nonlinear Anal., 2017], when the potential $b \in L^{\infty}(\mathbb{R}^{N})$ may change its sign, $0<a\in L^{\frac{2}{1+γ}}(\mathbb{R}^{N})$, $V$ is a positive continuous function, $N\geq 3$ and $λ>0$ is a real parameter. The main difficulties come from the non-differentiability of the energy functional…
Results on existence and multiplicity of solutions for a nonlinear elliptic problem driven by the $Φ$-Laplace operator are established. We employ minimization arguments on suitable Nehari manifolds to build a negative and a positive ground state solutions. In order to find a nodal solution we employ additionally the well known Deformation Lemma and Topological Degree Theory.
In this paper we study quasilinear elliptic equations driven by the double phase operator along with a reaction that has a singular and a parametric superlinear term and with a nonlinear Neumann boundary condition of critical growth. Based on a new equivalent norm for Musielak-Orlicz Sobolev spaces and the Nehari manifold along with the fibering method we prove the existence of at least two weak solutions provided the parameter is sufficiently small.
We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{array}{lr} -Δ_{p} u+a(x)|u|^{p-2}u=f(u)+ αλ(x)|u|^{α-2}u|v|^β, & x\in\mathbb{R}^{N}, -Δ_{q} v+b(x)|v|^{q-2}v=g(v)+ βλ(x)|v|^{β-2}v|u|^α, & x\in\mathbb{R}^{N}, \end{array} \right. \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $λ(x)$ of the coupling term is related with the potentials by the condition $|λ(x)|\leqδa(x)^{α/p}b(x)^{β/q}$ where $δ\in(0,1)$ and $α/p+β/q=1$. We deal with periodic and asymptotically periodic potentials. The nonlinear terms $f(s), \…
We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a $p$-Laplacian and of a weighted $q$-Laplacian ($q<p$) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter $λ>0$, the equation has at least two positive solutions.
We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*} M\left(\int_Ω|(-Δ)^{\fracα{2}}u|^2dx\right)(-Δ)^α u= λf(x)|u|^{q-2}u+|u|^{2^*_α-2}u\;\; \text{in}\; Ω,\;u=0\;\textrm{in}\;\mathbb R^n\setminus Ω, \end{equation*} where $Ω\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<α<1, \; 2α<n<4α$ and $ \; 1<q<2$. Here $2^*_α={2n}/{(n-2α)}$ is the fractional critical Sobolev exponent, $λ$ is a positive parameter and the coefficient $f(x)$ is…
In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: \begin{equation*}\begin{cases} -\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\dx\right)Δu+λV(x)u=\fracα{α+β}|u|^{α-2}u|v|^β,&x\in\mathbb{R}^3,\\ -\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2\dx\right)Δv+λW(x)v=\fracβ{α+β}|u|^α|v|^{β-2}v,&x\in\mathbb{R}^3,\\ u,v\in \mathcal{D}^{1,2}(\R^3), \end{cases}\end{equation*} where $a_i>0$ are constants, $λ,b_i>0$ are parameters for $i=1,2$, $α,β>1$ and $α+β\leqslant 4$, $V(x)$, $W(x)$ are nonnegative continuous potentials, the nonlinear term $F(x,u,v…
