Three solutions for quasilinear equations in \(\mathbb{R}^n\) near resonance. (English) Zbl 1014.35027

The authors study the following quasilinear elliptic equation in \(\mathbb{R}^n\): \[ \Delta_p u=(\lambda_1- \varepsilon)q(x) |u|^{p-2} u+f(x,u) +h(x), \tag{1} \] where \(\varepsilon >0\) small enough, \(\lambda_1\) is the first eigenvalue of \[ \begin{cases} -\Delta_pu=q(x) |u|^{p-2} u,\;x\in\mathbb{R}^n,\\ 0<u \text{ in }\mathbb{R}^n,\;\lim_{|x|\to +\infty}u(x)=0, \\ (1<p<n),\end{cases} \] \(g\) is a smooth function (at least \(C^{0,\gamma}_{\text{loc}} (\mathbb{R}^n)\) for some \(\gamma \in(0,1))\), \(f:\mathbb{R}^n\times\mathbb{R}\to \mathbb{R}\) is continuous and satisfies some growth condition, and \(h\in L^{p*'}\) with \(\int_{\mathbb{R}^n} h\varphi_1 dx=0\). Under some natural assumptions on the function \(f\) and the weight \(g\) they prove the existence of at least three solutions for (1) near resonance using minimax methods.
Reviewer: A.Krause (Berlin)


35J60 Nonlinear elliptic equations
35B34 Resonance in context of PDEs
35J20 Variational methods for second-order elliptic equations
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