相关论文: Mountain Pass Solutions for an entire semipositone…

For $N \geq 5$ and $a>0$, we consider the following semipositone problem \begin{align*} \Delta^2 u= g(x)f_a(u) \text { in } \mathbb{R}^N, \, \text{ and } \, u \in \mathcal{D}^{2,2}(\mathbb{R}^N),\ \ \ \qquad \quad \mathrm{(SP)} \end{align*}…

Let $s \in (0,1)$ and $N >2s$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-\Delta)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the…

For $N \geq 1, s\in (0,1)$, and $p \in (1, \frac{N}{s})$ we find a positive solution to the following class of semipositone problems associated with the fractional $p$-Laplace operator: \begin{equation}\tag{SP} (-\Delta)_{p}^{s}u =…

This article focuses on establishing a positive weak solution to a class of semipositone problems over the Heisenberg group $\mathbb{H}^N$. In particular, we are interested in the positive weak solution to the following problem:…

We consider the following Choquard equation $$ -\Delta_\gamma u + u = \left(d(z)^{-\mu} \ast |u|^p\right)|u|^{p-2}u, \text{ in } \mathbb{R}^N, $$ where $\Delta_\gamma$ is the Grushin operator. For a suitable range of the parameter $p$ we…

In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -\Delta _{p(\cdot )}u=f(u)-\lambda & \text{in }\Omega \\ u>0 &…

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-\Delta)_p^s(u) &=& \lambda f(u) \qquad & \text{in} \ \ \Omega \\u &=& 0 & \text{in} \…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-\Delta…

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -\Delta_\gamma u + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $\Delta_\gamma $ denotes the so-called…

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations $$ -\Delta_{\gamma} u + a(z)u = f(u) ~~ \mbox{in} ~~ \mathbb{R}^{N}, $$ where $\Delta_{\gamma}$ is known as the…

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: \begin{eqnarray*} (P_g)\quad - \Delta_{\lambda} u + V(x) u = f(x,u)+g(x),\;\mbox{ in } \R^N,\; \end{eqnarray*} and…

In this article, we prove the existence of at least one positive solution for the mixed local-nonlocal semipositone problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u+ (-\Delta)^s_p u &= \lambda f(u) && \text{in } \Omega, u &= 0…

We consider the boundary value problem $$ \cases{ -\Delta_\gamma u = \lambda u + \left\vert u \right\vert^{2^*_\gamma-2}u &in $\Omega$\cr u = 0 &on $\partial\Omega$,\cr } $$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N…

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type $$ (P_\lambda^\mu)\left\{ \begin{array}{lll} (-\Delta)^s u&=& \lambda(u^{q}-1)+\mu u^r \mbox{ in } \Omega\\ u&>&0 \mbox{ in…

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some…

We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation (-\Delta)^{s} u = g(u) \mbox{ in } \mathbb{R}^{N}, where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$ is the…

Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…

Using the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \frac{d}{dt}\mathcal{L}_v(t,u(t),\dot u(t))=\mathcal{L}_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*}…

Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u - \lambda V(x) u = h(x) u^{p-1}$ for $2<p$. The functions $V$ and $h$ may have an indefinite sign and the linearized operator…