相关论文: Ground state alternative for p-Laplacian with pote…

We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -\Delta_1 u +g(u)|Du|=h(u)f & \text{in $\Omega$,} \\ u=0 & \text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$,…

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…

For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…

In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -\Delta_1u=\lambda f(u) &\hbox{in }\Omega\,;\\[2mm] u=0…

In this paper, we consider the elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u=g(x,v)\,\, \textnormal{in}\Omega, & \hbox{} -\Delta v=f(x,u)\,\,\textnormal{in}\Omega, & \hbox{} u=v=0\textnormal{on}\partial\Omega, &…

We study positive solutions to the steady state reaction diffusion systems of the form: \begin{equation} \left\{\begin{array}{ll} -\Delta u = \lambda f(v)+\mu h(u), & \Omega,\\ -\Delta v = \lambda g(u)+\mu q(v),& \Omega,\\ \frac{\partial…

We consider the eigenvalue problem for the fractional $p \& q-$Laplacian \begin{equation} \left\{\begin{aligned} (- \Delta)_p^{s}\, u + \mu(- \Delta)_q^{s}\, u+ |u|^{p-2}u+\mu|u|^{q-2}u=\lambda\ V(x)|u|^{p-2}u\quad & \text{in } \Omega\\…

We obtain optimal regularity in the Sobolev space $W_0^{1,\tau}(\Omega)$ for the unique solution of $$ -\Delta_m u=K(x)u^{-p} \mbox{in} \Omega, \quad u=0\mbox{on}\partial \Omega. $$ Here $\Omega\subset{\mathbb R}^N$ is a smooth and bounded…

We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-\Delta_p^Nu=1$ in $\Omega$, $u=0$ and $\tfrac{\partial…

We investigate the multiplicity and uniqueness of positive solutions for the superlinear singular $(p,q)$-Laplacian equation \begin{eqnarray*} \begin{cases} -\Delta_p u-\Delta_q u+a(x)u^{p-1}+b(x)u^{q-1}=f(x)u^{-\gamma}+\lambda…

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $\Omega$ and $u= 0$ on…

Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive…

In this paper, we investigate the minimization problem : $$ \inf_{ \displaystyle{\begin{array}{lll} u \in H_0^1(\Omega), v \in H_0^1(\Omega),\\ \quad \| u \|_{L^{q}} =1, \quad \| v \|_{L^{q}} = 1 \end{array}}} \left[ \frac{1}{2}…

Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is…

In this paper, we study the following coupled Choquard system in $\mathbb R^N$: $$\left\{\begin{align}&-\Delta u+A(x)u=\frac{2p}{p+q} \bigl(I_\alpha\ast |v|^q\bigr)|u|^{p-2}u,\\ &-\Delta…

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 +…

We study the existence of positive radially symmetric solution for the singular $p$-Laplacian Dirichlet problem, $-\bigtriangleup_p u =\lambda |u|^{p-2} u-\gamma u^{-\alpha}$ where $\lambda>0,\gamma>0$ and, $0<\alpha<1$, are parameters and…

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative…

We study if two different solutions of the $p$-Laplace equation $$\nabla\cdot(|\nabla u|^{p-2}\nabla u)=0,$$ where $1<p<\infty$, can coincide in an open subset of their common domain of definition. We obtain some partial results on this…

In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, with $n\geq 4$, let $a, b,…