A note on the number of nodal solutions of an elliptic equation with symmetry

We consider the semilinear problem $-\Delta u+\lambda u={\left|u\right|}^{p-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain and $2<p<2^1=2N/(N-2)$. We show that if $\Omega$ is invariant under a nontrivial orthogonal involution then, for $\lambda >0$ sufficiently large, the equivariant topology of $\Omega$ is related to the number of solutions which change sign exactly once.     

Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents in RN

Let $N\ge 3,0\le s<2,0\le \mu <{\left(\frac{N-2}{2}\right)}^2$ and $2^1\left(s\right)≔\frac{2(N-s)}{N-2}$ be the critical Sobolev–Hardy exponents. Via variational methods and the analytic technique, we prove the existence of a nontrivial solution to the singular semilinear problem $-\Delta u-\mu \frac{u}{{\left|x\right|}^2}+u=\frac{{\left|u\right|}^{2^1\left(s\right)-2}}{{\left|x\right|}^s}u+f\left(u\right),u\in H_r^1\left({\mathbb{R}}^N\right)$, for $N\ge 4,0\le \mu \le \bar{\mu }-1$ and suitable functions $f\left(u\right)$.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Quasilinear elliptic problems with critical exponents and Hardy terms

Let $\Omega \ni 0$ be an open bounded domain, $\Omega \subset R^N(N>p^2)$. We are concerned with the multiplicity of positive solutions of $-{\mathrm{\Delta }}_{p}u-\mu \frac{|u{|}^{p-2}u}{|x{|}^p}=\lambda |u{|}^{p-2}u+Q(x)|u{|}^{p^{*}-2}u,u\in W_0^{1,p}(\Omega )\text{,}$where $-{\mathrm{\Delta }}_{p}u=-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u),1<p<N,p^{*}=\frac{\mathit{Np}}{N-p},0<\mu <{\left(\frac{N-p}{p}\right)}^p,\lambda >0$and $Q(x)$ is a nonnegative function on $\overline{\Omega }$. By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.     

Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition

In this paper, we study the following quasilinear Schrödinger equation $-\Delta u+u-\Delta \left(u^2\right)u=h\left(u\right),x\in {\mathbb{R}}^N,$where $N\ge 3$, $2^1=\frac{2N}{N-2}$, $h$ is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^4}=\infty$, we only need to assume that ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^2}=\infty$.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.