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Register a new userRobin double-phase problems with singular and superlinear terms
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We consider a nonlinear Robin problem driven by the sum of $p$-Laplacian and $q$-Laplacian (i.e. the $(p,q)$-equation). In the reaction there are competing effects of a singular term and a parametric perturbation $lambda f(z,x)$, which is Caratheodory and $(p-1)$-superlinear at $xinmathbb{R},$ without satisfying the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $lambda>0$ varies.
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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 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 $lambda>0$, the equation has at least two positive solutions.
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
In this paper we study quasilinear elliptic equations driven by the double phase operator and a right-hand side which has the combined effect of a singular and of a parametric term. Based on the Nehari manifold method we are going to prove the existence of at least two weak solutions for such problem when the parameter is sufficiently small.
We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: begin{equation*} -left( aint_{mathbb{R}^{N}}| abla u|^{2}dx+1right) Delta u+mu V(x)u=lambda f(x)u+g(x)|u|^{p-2}uquad text{ in }mathbb{R}^{N}, end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2}$, $Vin C(mathbb{R}^{N})$ is a potential well with the bottom $Omega :=int{xin mathbb{R}^{N} | V(x)=0}$. When $N=3$ and $4<p<6$, for each $a>0$ and $mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0<lambdaleqlambda _{1}(f_{Omega}) $ while at least two positive solutions exist for $lambda _{1}(f_{Omega })< lambda<lambda _{1}(f_{Omega})+delta_{a}$ without any assumption on the integral $% int_{Omega }g(x)phi _{1}^{p}dx$, where $lambda _{1}(f_{Omega })>0$ is the principal eigenvalue of $-Delta $ in $H_{0}^{1}(Omega )$ with weight function $f_{Omega }:=f|_{Omega }$, and $phi _{1}>0$ is the corresponding principal eigenfunction. When $Ngeq 3$ and $2<p<min {4,2^{ast }}$, for $% mu $ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<lambda <lambda _{1}(f_{Omega })$; $% (ii)$ under the classical assumption $int_{Omega }g(x)phi _{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $lambda _{1}(f_{Omega })leq lambda<lambda _{1}(f_{Omega})+overline{delta }% _{a} $; $(iii)$ under the assumption $int_{Omega }g(x)phi _{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $lambda^{+}_{a}< lambda<lambda _{1}(f_{Omega})$ for some $a_{0}(p)>0$ and $lambda^{+}_{a}geq0$.
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