Related papers: On a class of parametric $(p,2)$-equations

We consider Dirichlet elliptic equations driven by the sum of a $p$-Laplacian $(2<p)$ and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both $\pm\infty$ and at zero. We prove an existence theorem…

We consider a $(p,2)$-equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a $p$-Laplacian and a Laplacian with $p>2$. The reaction term is $(p-1)$-linear but exhibits asymmetric behaviour at $\pm\infty$ and…

In this paper we study an eigenvalue problem for the so called $(p,2)$-Laplace operator on a smooth bounded domain under a nonlinear Steklov type boundary condition, namely \begin{equation} \left\{ \begin{aligned} -\Delta_pu-\Delta u &…

The main purpose of this paper is to show that there exists a positive number $\lambda_{1}$, the first eigenvalue, such that some $p(x)$-Laplace equation admits a solution if $\lambda=\lambda_{1}$ and that $\lambda_{1}$ is simple, i.e.,…

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\Omega \subset \mathbb{R}^N$. By means of…

We study the behavior of solutions for the parametric equation $$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$ under Dirichlet condition, where $\Omega \subseteq…

For a class of quasilinear elliptic equations involving the p-Laplace operator, we develop an abstract critical point theory in the presence of sub-supersolutions. Our approach is based upon the proof of the invariance under the gradient…

We derive explicit ground state solutions for several equations with the $p$-Laplacian in $R^n$, including (here $\varphi (z)=z|z|^{p-2}$, with $p>1$) \[ \varphi \left(u'(r)\right)' +\frac{n-1}{r} \varphi \left(u'(r)\right)+u^M+u^Q=0 \,. \]…

We consider a nonlinear parametric Dirichlet problem driven by the $p$-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carath\'eodory perturbation which is ($p-1$)-linear…

A Dirichlet problem driven by the $(p,q)$-Laplace operator and an asymmetric concave reaction with positive parameter is investigated. Four nontrivial smooth solutions (two positive, one negative, and the remaining nodal) are obtained once…

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently…

We study the global solution curves, and prove the existence of infinitely many positive solutions for three classes of self-similar equations, with $p$-Laplace operator. In case $p=2$, these are well-known problems involving the Gelfand…

Let $1<p<N$, $p^{*}=Np/(N-p)$, $0<s<p$, $p^{*}(s)=(N-s)p/(N-p)$, and $\Om\in C^{1}$ be a bounded domain in $\R^{N}$ with $0\in\bar{\Om}.$ In this paper, we study the following problem \[ \begin{cases}…

In this article, we use results of Number Theory to prove the conjecture on eigenvalue problem of a 2D elliptic PDE proposed by P. Korman in his recent paper \cite{ref}: for any even integer $2k$, one can find an eigenvalue $N$ that can be…

We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the $p$-Laplacian. The natural framework in which we consider such equations is that of Orlicz-Sobolev spaces. we exhibit two…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

We investigate the existence and multiplicity of abstract weak solutions of the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and…

Using a variational technique we guarantee the existence of a solution to the \emph{resonant Lane-Emden} problem $-\Delta_p u=\lambda |u|^{q-2}u$, $u|_{\partial\Omega}=0$ if and only if a solution to $-\Delta_p u=\lambda |u|^{q-2}u+f$,…

In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p} u -\Delta_{q} u +V(\varepsilon x) \left(|u|^{p-2}u +…

Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p \Delta _{p}u + u^{p-1} = f(u) & \hbox{in} \ \Omega \newline u>0 & \hbox{in} \ \Omega \newline u=0 & \hbox{on} \ \partial \Omega…