On resonant \((p,2)\)-equations (Q2789198)

Let \(\Omega\subset \mathbb{R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial\Omega\)\,. In this paper, the authors study the following nonlinear nonhomogeneous elliptic equation: NEWLINE\[NEWLINE -\Delta_p u(z) -\Delta u(z) = f(z,u(z))\,\; \text{in}\;\Omega,\; u_{|\partial\Omega}=0\quad (2<p<\infty)\,. \eqno{(1)} NEWLINE\]NEWLINENEWLINENEWLINETo formulate main results, the authors denote the positive cone NEWLINE\[NEWLINE C_+ = \{u \in C^1_0 (\overline{\Omega}) : u(z)\geq 0 \; \text{for all} \; z\in \overline{\Omega} \},NEWLINE\]NEWLINE its interior NEWLINE\[NEWLINE \text{int}C_+ =\{u \in C_+ : u(z)> 0 \; \text{for all} \; z\in\Omega, \frac{\partial u}{\partial n} <0 \; \text{for all} \; z\in\partial \Omega\} NEWLINE\]NEWLINE and a smallest eigenvalue NEWLINE\[NEWLINE \hat{\lambda}_1(p) = \inf \left[\frac{\|Du\|_p^p}{\|u\|_p^p}: u \in W^{1,p}_0 (\Omega), u \not= 0\right]. NEWLINE\]NEWLINENEWLINENEWLINEFirstly, the authors produce solutions of constant sign for problem (1). To this end, they introduce the following hypotheses on the reaction \(f(z, x)\). \(\quad\)NEWLINENEWLINE\(H_4\): The function \(f : \Omega\times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function s.t. \(f(z, 0) = 0\) for a.a. \(z\in\Omega\), and {\parindent=6mm \begin{itemize} \item[(i)] \(|f(z, x)| \leq a(z)(1 + |x|^{p-1})\) for a.a. \(z\in\Omega\), all \(x \in \mathbb{R}\) , with \(a \in L^\infty(\Omega)_+\); \item [(ii)] \(\limsup_{x\to \pm\infty}\frac{f(z,x)}{|x|^{p-2}x} \leq \hat{\lambda}_1(p)\) uniformly for a.a. \(z\in\Omega\) and there exists \(\xi_0 > 0\) s.t. NEWLINE\[NEWLINEf(z, x)x - pF(z, x) \geq -\xi_0\quad\text{for a.a}\;\, z\in\Omega,\text{all}\;\, x \in\mathbb{R};NEWLINE\]NEWLINE \item [(iii)] there exist functions \(\eta, \hat{\eta} \in L^\infty(\Omega)_+\), s.t. NEWLINE\[NEWLINE\hat{\lambda}_2(2)< \eta(z) \leq \hat{\eta}(z) \quad\text{for a.a.}\;\, z\in\Omega, NEWLINE\]NEWLINE NEWLINE\[NEWLINE\eta (z) \leq \liminf_{x\to 0}\frac{f(z, x)}{x} \leq \limsup_{x\to 0}\frac{f(z, x)}{x} \leq \hat{\eta}(z) \quad\text{uniformly for a.a.}\;\, z\in\Omega;NEWLINE\]NEWLINE \item [(iv)] for every \(\varrho > 0\) there exists \(\hat{\xi}_{\varrho} > 0\) s.t. for a.a. \(z\in\Omega\), NEWLINE\[NEWLINEx \mapsto f(z, x) + \hat{\xi}_{\varrho}|x|^{p-2}xNEWLINE\]NEWLINE is non decreasing on \([-\varrho , \varrho ].\) NEWLINENEWLINE\end{itemize}} The main result of this paper is the following theorem.NEWLINENEWLINE{ Theorem 4.4.} If hypotheses \(H_4\) hold, then problem (1) has at least three nontrivial solutions \(u_0 \in \text{int} C_+, v_0 \in -\text{int} C_+\) and \(y_0 \in \text{int}_{C^1_0(\Omega)}[v_0, u_0]\) nodal.NEWLINENEWLINEThe authors also improve this theorem and produce additional solutions for problem (1). The new hypotheses on \(f(z, x)\) are the following:NEWLINENEWLINE\(H_5\): The function \(f : \Omega\times \mathbb{R} \to \mathbb{R}\) is a measurable function s.t. \(f(z, 0) = 0\) for a.a. \(z\in\Omega\), \(f(z, 0) = 0, f(z,\cdot ) \in C^1(R)\) and {\parindent=6mm \begin{itemize} \item[(i)] \(|f_x'(z, x)| \leq a(z)(1 + |x|^{p-2})\) for a.a. \(z\in\Omega\), all \(x \in \mathbb{R}\) , with \(a \in L^\infty(\Omega)_+\); \item [(ii)] \(\limsup_{x\to \pm\infty}\frac{f(z,x)}{|x|^{p-2}x} \leq \hat{\lambda}_1(p)\) uniformly for a.a. \(z\in\Omega\) and there exists \(\xi_0 > 0\) s.t. NEWLINE\[NEWLINEf(z, x)x - pF(z, x) \geq -\xi_0\quad\text{for a.a}\;\, z\in\Omega,\text{all}\;\, x \in \mathbb{R};NEWLINE\]NEWLINE \item [(iii)] \(f'_x(z, 0) = \lim_{x\to 0}\frac{f(z,x)}{x}\) uniformly for a.a. \(z\in\Omega\) and there exists an integer \(m\geq 2\) s.t. NEWLINE\[NEWLINEf'_x(z, 0) \in [\hat{\lambda}_m(2),\hat{\lambda}_{m+1}(2)]\quad\text{for a.a.}\;\, z\in\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf'_x(\cdot, 0) \not\equiv \hat{\lambda}_m(2), \, f'_x(\cdot, 0) \not\equiv \hat{\lambda}_{m+1}(2) .NEWLINE\]NEWLINE NEWLINENEWLINE\end{itemize}} They state the second multiplicity theorem for problem (1).NEWLINENEWLINE{ Theorem 4.8.} If hypotheses \(H_5\) hold, then problem (1) admits at least four nontrivial solutions NEWLINE\[NEWLINE u_* \in \text{int} C_+, v_* \in -\text{int} C_+ \; \text{and} \; y_0, \hat{y} \in \text{int}_{C^1_0 (\overline{\Omega})}[v_*, u_*]\text{ nodal}.NEWLINE\]NEWLINE The authors also allow resonance at zero and obtain a similar multiplicity theorem producing four nontrivial solutions, but in this case they can not say that the fourth solution is nodal.NEWLINENEWLINETo prove these theorems, the authors use minimax methods based on the critical point theory, together with truncation techniques and Morse theory.