In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
This paper is concerned with the following planar Schrodinger-Poisson system: $\begin{cases} -\Delta u +V(x)u+\lambda\phi u=u\ln u^2,&\text{in }{R}^2,\\ \Delta\phi=u^2,&\text{in }{R}^2. \end{cases}$ where $\lambda\in{R}$ is a parameter and $V\in C^1({R}^2,R}^{+})$ is a coercive potential. Due to the presence of the log...
Let $1 \leq q<p$ and let $\lambda_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $\Omega$. We study the energy functional $$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\,dx -\lambda\int_\Omega|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ where $\mathcal{F}$ is positive...
In a Lipschitz domain
$$\Omega \subset \mathbb {R}^N$$
, we investigate the boundary behavior of positive superharmonic functions
u
satisfying the nonlinear inequality
$$-\Delta u(x)\le cd(x,\partial \Omega )^{-\tau }u(x)^p$$
. Based on the fact that, in some ranges of
p
and
$$\tau $$
, the po...
Kentaro Hirata· The Journal of Analysis· 0 citations
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harm...
We study positive solutions of the partially confined stationary nonlinear Schr\"odinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-\Delta + |y|^2$ is given by $[d,\infty)$. We prove that $\lambda\geq-d$ is necessary for po...
Zhijie Chen, Youjun Wang, Jian-Jun Zhang et al.· 0 citations
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