Related papers: Multiplicity results in the non-coercive case for …

We consider the boundary value problem $-\Delta_p u = \lambda c(x) |u|^{p-2}u + \mu(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(\Omega) \cap L^{\infty}(\Omega)$, where $\Omega \subset \mathbb R^N$, $N \geq 2$, is a bounded domain with smooth…

We consider the boundary value problem \begin{equation*} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega) \eqno{(P_{\lambda})} \end{equation*} where $\Omega \subset \R^N, N \geq 3$ is…

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_{\lambda}$} -\Delta u = c_{\lambda}(x) u + \mu |\nabla u|^2 + h(x)\,,…

Let $\Omega \subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-\Delta u = c_{\lambda}(x) u + \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega)\cap L^{\infty}(\Omega)$$ where…

We consider the following problem $$(P) \begin{cases} -\Delta_{p}u= c(x)|u|^{q-1}u+\mu |\nabla u|^{p}+h(x) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega, \end{cases}$$ where $\Omega$ is a bounded set in $\mathbb{R}^{N}$…

We consider the problem $(P)$, $$ -\Delta u =c(x)u+\mu|\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega),$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $\mu>0, \, c \in…

We study nonnegative solutions of the boundary value problem $$-\Delta u = \lambda c(x)u + \mu(x)|\nabla u|^2 + h(x),\quad u\in H^1_0(\Omega)\cap L^\infty(\Omega), \leqno(P_\lambda)$$ where $\Omega$ is a smooth bounded domain, $\mu, c\in…

Given $\mu > 0$, we study the elliptic problem: \begin{align*} \text{ find } (u,\lambda) \in H_0^1(\Omega) \times \mathbb{R} \text{ such that } -\Delta u + \lambda u = |u|^{p-2}u \text{ in } \Omega \text{ and } \int_\Omega|u|^2dx = \mu,…

In this paper, we prove a theorem concerning the existence of three solutions for the following boundary value problem: \begin{equation*} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u)-\Gamma|Du|^2=f(u)~~~\text{in}\ \Omega, u=0~~~\text{on}\…

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u…

We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_\lambda$} -\mathrm{div}(A(x)Du)=c_\lambda(x)u+( M(x)Du,Du)+h(x),\qquad u\in…

In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= \lambda u^{q} + u^{p}, \quad u>0 ~~ \text{in} ~\Omega,…

In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \frac{\lambda}{u^{\gamma}}+g(u)+\mu~\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\,…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x…

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-\Delta_p)^s u&=& \frac{\lambda}{u^\gamma}+u^q~\text{in}~\Omega,\nonumber…

We will prove multiplicity results for the mixed local-nonlocal elliptic equation of the form \begin{eqnarray} \begin{split} -\Delta_pu+(-\Delta)_p^s u&=\frac{\lambda}{u^{\gamma}}+u^r \text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0…

The boundary value problem is examined for the system of elliptic equations of from $-\Delta u + A(x)u = 0 \quad\text{in} \Omega,$ where $A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ and $\frac{\partial u}{\partial…

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{cases} -\Delta_p u = H(u)\mu & \text{in}\ \Omega,\\ u>0…